Modelling of Mechanical Systems: Fluid-Structure Interaction, Volume 3 (Modelling of Mechanical Systems)

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MODELLING OF MECHANICAL SYSTEMS

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MODELLING OF MECHANICAL SYSTEMS: FLUID STRUCTURE INTERACTION Volume 3 François Axisa and Jose Antunes

Butterworth-Heinemann is an imprint of Elsevier

Butterworth-Heinemann is an imprint of Elsevier Linacre House, Jordan Hill, Oxford OX2 8DP, UK 30 Corporate Drive, Suite 400, Burlington, MA 01803, USA First edition 2007 Copyright © 2007, François Axisa and Jose Antunes. Published by Elsevier Ltd. All rights reserved. The right of François Axisa and Jose Antunes to be identified as the authors of this work has been asserted in accordance with the Copyright, Designs and Patents Act 1988 No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means electronic, mechanical, photocopying, recording or otherwise without the prior written permission of the publisher Permissions may be sought directly from Elsevier's Science & Technology Rights Department in Oxford, UK: phone (+44) (0) 1865 843830; fax (+44) (0) 1865 853333; email: [email protected]. Alternatively you can submit your request online by visiting the Elsevier web site at http://elsevier.com/locate/permissions, and selecting Obtaining permission to use Elsevier material Notice No responsibility is assumed by the publisher for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein. Because of rapid advances in the medical sciences, in particular, independent verification of diagnoses and drug dosages should be made British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library Library of Congress Cataloging in Publication Data A catalog record for this book is available from the Library of Congress

ISBN-13: 978-0-750-66847-7 ISBN-10: 0-7506-6847-4

For information on all Butterworth-Heinemann publications visit our website at http://books.elsevier.com Printed and bound in Great Britain 07 08 09 10

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Contents

Preface ..................................................................................................................... xv Introduction .......................................................................................................... xvii Chapter 1. Introduction to fluid-structure coupling ............................................. 1 1.1. A short outline of fluid-structure coupled systems ............................................ 2 1.1.1. Basic mechanism of fluid-structure dynamical coupling ...................... 2 1.1.2. A few elementary experiments.............................................................. 4 1.2. Dynamic equations of fluid-structure coupled systems ................................... 10 1.2.1. Elastic vibrations of solid structures ................................................... 10 1.2.2. Dynamic equations of Newtonian fluids............................................. 12 1.2.2.1. Eulerian acceleration and material derivative...................... 12 1.2.2.2. Mass-conservation equation ................................................ 13 1.2.2.3. Momentum equation............................................................ 14 1.2.2.4. Pressure and fluid elasticity ................................................. 15 1.2.2.5. Fluid elasticity and equation of state of a gas ...................... 18 1.2.2.6. Cavitation of a liquid ........................................................... 20 1.2.2.7. Viscous stresses ................................................................... 21 1.2.2.8. Navier-Stokes equations ...................................................... 23 1.3. Linear approximation of the fluid equations.................................................... 26 1.3.1. Linearized fluid equations about a quiescent state .............................. 26 1.3.1.1. Linear Navier-Stokes equations........................................... 26 1.3.1.2. The linear Euler equations ................................................... 27 1.3.1.3. The sound wave equation in terms of a single field............. 27 1.3.2. Linearized boundary conditions .......................................................... 28 1.3.2.1. Fluid-structure coupling term at a wetted wall .................... 28 1.3.2.2. Free surface of a liquid in a gravity field............................. 29 1.3.2.3. Surface tension at the interface between two fluids............. 34 1.3.3. Physical quantities and oscillations of the fluid .................................. 39 1.3.3.1. Mean value of fluid density ................................................. 39

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1.3.3.2. 1.3.3.3. 1.3.3.4. 1.3.3.5.

Gravity field......................................................................... 41 Surface tension .................................................................... 41 Fluid elasticity ..................................................................... 42 Fluid viscosity...................................................................... 43

Chapter 2. Inertial coupling .................................................................................. 45 2.1. Introduction ..................................................................................................... 46 2.2. Discrete systems .............................................................................................. 48 2.2.1. The fluid column model ...................................................................... 48 2.2.2. Single degree of freedom systems....................................................... 51 2.2.2.1. Piston-fluid system: tube of uniform cross-section.............. 51 2.2.2.2. Piston-fluid system as a dynamically coupled system ......... 52 2.2.2.3. Piston-fluid system: tube of variable cross-section.............. 55 2.2.2.4. Hole and inertial impedance ................................................ 57 2.2.2.5. Response to a seismic excitation ......................................... 59 2.2.2.6. Nonlinear inertia in piping systems ..................................... 66 2.2.3. Systems with spherical symmetry ....................................................... 68 2.2.3.1. Breathing mode of a spherical shell immersed in a liquid................................................................................................ 68 2.2.3.2. Early stage of a submarine explosion .................................. 71 2.2.4. Piston-fluid system with two degrees of freedom ............................... 76 2.2.4.1. Natural modes of vibration .................................................. 76 2.2.4.2. Lagrange’s equations ........................................................... 78 2.2.4.3. Newtonian treatment of the problem ................................... 80 2.3. Continuous systems ......................................................................................... 81 2.3.1. Modal added mass matrix ................................................................... 81 2.3.2. Strip model of elongated fluid-structure systems................................ 84 2.3.2.1. Cylindrical shells of revolution............................................ 84 2.3.2.2. Cylindrical shell immersed in an infinite extent of liquid .............................................................................................. 92 2.3.2.3. Inertial coupling of two coaxial circular cylindrical shells................................................................................................... 94 2.3.3. Thin fluid layer approximation ........................................................... 98 2.3.3.1. Concentric cylindrical shells of revolution .......................... 98 2.3.3.2. Extension to other geometries............................................ 100 2.3.4. Mode shapes modified by fluid inertia.............................................. 103 2.3.4.1. Rigid rod partly immersed in a liquid ................................ 104 2.3.4.2. Coaxial cylindrical shells of revolution ............................. 107 2.3.4.3. Water tank with flexible lateral walls ................................ 114 2.3.5. 3D problems...................................................................................... 120 2.3.5.1. Plate immersed in a liquid layer of finite depth ................. 120 2.3.5.2. Circular cylindrical shell of low aspect ratio ..................... 122 2.3.5.3. Vertical oscillation of an immersed spherical object ......... 129 2.3.5.4. The immersed sphere used as an inverted pendulum......... 131

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Chapter 3. Surface waves..................................................................................... 138 3.1. Introduction ................................................................................................... 139 3.2. Gravity waves................................................................................................ 140 3.2.1. Harmonic waves in a rectilinear canal .............................................. 140 3.2.2. Group velocity and propagation of wave energy .............................. 145 3.2.3. Shallow water waves ( kH > 1) ........................................................... 158 3.2.5.1. Space and time profiles of progressive waves ................... 159 3.2.5.2. Wake of a moving boat and Kelvin wedge........................ 166 3.2.6. Water waves at intermediate depths: solitary waves......................... 168 3.2.7. Wave impacting a rigid wall ............................................................. 172 3.3. Surface tension .............................................................................................. 179 3.3.1. Capillary waves, or ripples................................................................ 179 3.3.2. Surface tension and cavitation .......................................................... 181 3.3.2.1. Static equilibrium of a micro-bubble, or cavitation nucleus.............................................................................................. 181 3.3.2.2. The collapse of cavitation bubbles..................................... 186 3.3.2.3. Oscillations and activation of the cavitation nuclei ........... 189 3.3.2.4. Rayleigh-Plesset equation.................................................. 192 3.4. Sloshing modes.............................................................................................. 205 3.4.1. Discrete systems................................................................................ 205 3.4.1.1. U tube ................................................................................ 205 3.4.1.2. Interconnected tanks .......................................................... 207 3.4.2. Continuous systems........................................................................... 209 3.4.2.1. Rectangular tank ................................................................ 209 3.4.2.2. Circular tank ...................................................................... 213 3.5. Fluid-structure interaction ............................................................................. 216 3.5.1. Coupling between sloshing and structural modes ............................. 216 3.5.2. Floating structures............................................................................. 224 3.5.2.1. Introduction ....................................................................... 224 3.5.2.2. Buoyancy of a boat ............................................................ 225 3.5.2.3. Stability of the static equilibrium....................................... 226 3.5.2.4. Natural frequencies of the rigid body modes ..................... 228 3.5.2.5. Example 1: heave mode of a floating circular cylindrical buoy ................................................................................ 230 3.5.2.6. Example 2: rectangular cross-section ................................ 233 3.5.2.7. Rolling induced by the swell ............................................. 238 3.5.2.8. Antiresonant absorber for rolling....................................... 239

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Chapter 4. Plane acoustical waves in pipe systems ............................................ 243 4.1. Introduction ................................................................................................... 244 4.1.1. Acoustics and sound perception........................................................ 244 4.1.2. Acoustics in the context of fluid-structure interaction ...................... 245 4.1.3. Linear and conservative acoustical wave equation ........................... 245 4.2. Free sound waves in pipe systems: plane and harmonic waves..................... 247 4.2.1. Acoustic impedances and standing sound waves .............................. 247 4.2.1.1. Plane wave approximation in pipes ................................... 247 4.2.1.2. Plane wave equations in pipes ........................................... 248 4.2.1.3. Travelling waves in a uniform tube and tube impedance......................................................................................... 249 4.2.1.4. Reflected and transmitted waves at a change of impedance......................................................................................... 250 4.2.1.5. Reflected and transmitted waves through three media ...... 252 4.2.1.6. Boundary conditions and terminal impedances ................. 256 4.2.1.7. Radiation damping and complex impedance ..................... 261 4.2.1.8. Acoustical modes in a uniform tube .................................. 262 4.2.1.9. Application to wind musical instruments .......................... 267 4.2.1.10. Horns: Webster and Schrödinger equations..................... 269 4.2.1.11. Bessel horns..................................................................... 272 4.2.2. Transfer matrix method (TMM)........................................................ 279 4.2.2.1. Transfer matrix of a uniform tube element ........................ 279 4.2.2.2. Assembling of two tube elements ...................................... 282 4.2.2.3. Two connected tubes of distinct cross-sectional areas....... 284 4.2.2.4. Two connected tubes filled with distinct fluids ................. 286 4.2.2.5. Helmholtz resonators ......................................................... 289 4.2.2.6. Higher plane wave modes of an enclosure tube assembly ........................................................................................... 293 4.2.2.7. Enclosure-tube assembly: case of a very short tube........... 295 4.3. Forced waves ................................................................................................. 296 4.3.1. Concentrated acoustical sources ....................................................... 296 4.3.1.1. Volume velocity (monopole) source.................................. 296 4.3.1.2. Pressure (dipole) source..................................................... 299 4.3.2. Transfer functions for a uniform tube ............................................... 299 4.3.2.1. Transfer matrix method ..................................................... 300 4.3.2.2. Modal expansion method................................................... 309 4.3.3. Acoustical isolation of a piping system............................................. 310 4.3.3.1. Cavity inserted in series with the main circuit................... 311 4.3.3.2. Cavity connected in derivation to the main circuit ............ 316 4.3.4. Computational procedures suited to TMM softwares ....................... 320 4.3.4.1. Formulation of the forced acoustical system ..................... 320 4.3.4.2. Matrix equation of a tube element ..................................... 321 4.3.4.3. Impedances and external sources....................................... 322 4.3.4.4. Single branched circuits..................................................... 324 4.3.4.5. Multi-branched circuits...................................................... 325

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4.3.4.6. Application to the acoustical isolation of a forced flow loop........................................................................................... 327 4.4. Speed of sound .............................................................................................. 329 4.4.1. Speed of sound and fluid compressibility ......................................... 329 4.4.2. Isothermal versus adiabatic speed of sound in gases......................... 334 4.4.3. Speed of sound in a gas liquid mixture (bubbly liquid) .................... 339 4.4.3.1. Quasi-static homogeneous model ...................................... 339 4.4.3.2. Dispersive model accounting for the bubble vibrations..... 345 4.4.4. Speed of sound of a fluid contained within elastic walls .................. 349 Chapter 5. 3D Sound waves ................................................................................. 353 5.1. 3D Standing sound waves (acoustic modes).................................................. 354 5.1.1. Modal equations and general properties of acoustic modes .............. 354 5.1.1.1. Interface separating two media and boundary conditions ......................................................................................... 354 5.1.1.2. Wave equation expressed in terms of displacement field................................................................................................... 358 5.1.1.3. Wave equation expressed in terms of pressure .................. 362 5.1.2. Analytical examples of acoustical modes ......................................... 364 5.1.2.1. Rectangular enclosure........................................................ 364 5.1.2.2. Circular cylindrical enclosure............................................ 369 5.1.2.3. Spherical enclosure............................................................ 378 5.2. Guided wave modes and plane wave approximation..................................... 387 5.2.1. Introduction....................................................................................... 387 5.2.2. Rectangular waveguides ................................................................... 388 5.2.2.1. Guided mode waves........................................................... 388 5.2.2.2. Physical interpretation ....................................................... 393 5.2.3. Cylindrical waveguides..................................................................... 396 5.3. Forced waves ................................................................................................. 398 5.3.1. Forced wave equations...................................................................... 398 5.3.2. Forced waves in rectangular enclosures............................................ 399 5.3.2.1. Green function ................................................................... 399 5.3.2.2. Response to a velocity source distributed over a surface............................................................................................ 401 5.3.2.3. Response to a concentrated pressure, or dipole source ..... 405 5.3.2.4. Modal expansion method for coupled enclosures.............. 406 5.3.3. Forced waves in waveguides............................................................. 411 5.3.3.1. Local and far acoustical fields ........................................... 412 5.3.3.2. Impedance surface and mode coupling.............................. 417 5.3.4. Forced waves in open space: Green’s functions ............................... 421 5.3.4.1. 3D unbounded medium...................................................... 421 5.3.4.2. 3D medium bounded by a fixed plane, image source method .............................................................................................. 424 5.3.4.3. 3D medium bounded by a pressure nodal plane, dipole sources ................................................................................... 426

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5.3.4.4. Distributed monopole sources and 2D cylindrical waves ................................................................................................ 427 5.3.4.5. Distributed monopole sources and plane waves ................ 429 5.3.4.6. Distributed monopole sources and first Rayleigh integral.............................................................................................. 431 5.3.4.7. Pressure field in the axial direction by a baffled circular piston ................................................................................... 432 5.3.4.8. Directivity of sound radiated by a baffled circular piston ................................................................................................ 436 5.3.4.9. Dipole radiation by the unbaffled circular piston integral equation (KH)...................................................................... 437 5.3.5. Weighted integral formulations......................................................... 439 5.3.5.1. The Kirchhoff-Helmholtz integral theorem ....................... 439 5.3.5.2. Particularization of K.H. integral to plane waves .............. 441 5.3.5.3. Application to plane acoustic waves triggered by a transient ......................................................................................... 446 5.3.5.4. K.H. integral for 3D external and internal problems ......... 452 5.3.5.5. Application: pressure field induced by the unbaffled circular piston ................................................................................... 457 Chapter 6. Vibroacoustic coupling...................................................................... 461 6.1. Local equilibrium equations .......................................................................... 462 6.1.1. Mixed and non symmetrical formulation .......................................... 462 6.1.2. Symmetrical formulation in terms of displacements......................... 463 6.1.3. Mixed and symmetrical formulation ................................................. 464 6.2. Piston-fluid column system ........................................................................... 465 6.2.1. Modal problem.................................................................................. 466 6.2.1.1. Analytical solution............................................................. 466 6.2.1.2. Modal expansion method: displacement as the fluid variable ............................................................................................. 471 6.2.1.3. Modal expansion method: pressure as the fluid variable ... 477 6.2.1.4. Pressure and displacement potential as two fluid variables............................................................................................ 481 6.2.2. Analytical solutions of forced problems ........................................... 486 6.2.2.1. Piston coupled to a fluid column and forced harmonically ..................................................................................... 486 6.2.2.2. Response to a transient force exerted on the piston ........... 490 6.2.2.3. Tube excited by a transient pressure source ...................... 493 6.2.3. Expansion methods to solve forced problems................................... 497 6.2.3.1. Displacement field as the fluid variable............................. 497 6.2.3.2. Response to a seismic excitation ....................................... 502 6.3. Vibroacoustic coupling in tube and ducts circuits ......................................... 506 6.3.1. Simplifications inherent in the tubular geometry .............................. 506 6.3.2. Tubular vibroacoustic coupling model.............................................. 507 6.3.2.1. Incompressible transverse coupling terms ......................... 508

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6.3.2.2. Vibroacoustic coupling at a change in the cross-section ... 510 6.3.2.3. Vibroacoustic coupling at bends........................................ 512 6.3.2.4. Vibroacoustic coupling at closed ends and tube junctions ........................................................................................... 514 6.3.2.5. Equation of motion of a pipe filled with a fluid................. 515 6.4. Application to a few problems....................................................................... 517 6.4.1. Vibroacoustic modes of cylindrical vessels ...................................... 518 6.4.1.1. Longitudinal vibroacoustic modes of a straight vessel ...... 518 6.4.1.2. Numerical aspects related to the modal projection method .............................................................................................. 524 6.4.1.3. Vibroacoustic modes of an inflated toroidal shell ............. 531 6.4.1.4. Thermal expansion lyre filled with incompressible fluid .. 539 6.4.1.5. Thermal expansion lyre filled with compressible fluid...... 543 6.4.2. Simplified model of a drum using modal expansions ....................... 548 6.4.3. Vibroacoustic consequences of cavitation ........................................ 554 6.4.3.1. One dimensional model of cavitation ................................ 554 6.4.3.2. Analytical example ............................................................ 556 6.5. Finite element method ................................................................................... 563 6.5.1. Introduction....................................................................................... 563 6.5.2. Variational formulation of the vibroacoustic equations .................... 564 6.5.2.1. Formulation in terms of fluid displacement....................... 564 6.5.2.2. Mixed ( X S , p ) formulation.............................................. 568 6.5.2.3. Mixed ( X s , Π , p ) formulation ......................................... 569 6.5.3. Discretization in finite elements........................................................ 571 6.5.3.1. Finite element equations in the ( X s , X f ) variables .......... 573 6.5.3.2. Finite element equations in the ( X s , p ) variables............. 575 6.5.3.3. Finite element equations in the ( X s , Π , p ) variables....... 578 6.5.3.4. Example: 1D acoustic finite element ................................. 578 Chapter 7. Energy dissipation by the fluid......................................................... 581 7.1. Preliminary survey on linear modelisation of dissipation.............................. 582 7.1.1. Diversity and importance of the dissipative processes...................... 582 7.1.2. The viscous damping model.............................................................. 582 7.1.2.1. Damped harmonic oscillator.............................................. 583 7.1.2.2. Multiple degrees of freedom systems ................................ 585 7.1.2.3. Damped acoustical modes in a tube................................... 595 7.1.2.4. Transfer matrix method ..................................................... 599 7.1.3. Forced damped waves....................................................................... 604 7.1.3.1. Spectral domain ................................................................. 604 7.1.3.2. Time domain: dissipative terminal impedance .................. 607 7.1.3.3. Time domain: dissipative fluid .......................................... 608

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7.2. Radiation damping......................................................................................... 613 7.2.1. Radiation of acoustic waves.............................................................. 613 7.2.1.1. Sound intensity and power levels ...................................... 613 7.2.1.2. Piston-fluid column system: motion of the piston ............. 616 7.2.1.3. Piston-fluid column system: acoustic waves ..................... 618 7.2.1.4. Piston-fluid system: terminal impedance for an open tube .......................................................................................... 624 7.2.1.5. Spherical shell pulsating in an infinite medium................. 629 7.2.1.6. Kirchhoff-Helmholtz integral applied to the spherical radiator.............................................................................................. 638 7.2.1.7. Rigid sphere oscillating rectilinearly ................................. 643 7.2.1.8. Radiation of circular cylindrical shells .............................. 647 7.2.2. Sound transmission through interfaces ............................................. 651 7.2.2.1. Transmission loss at the interface separating two fluids.... 651 7.2.2.2. Transmission through a flexible wall: “infinite” and “finite” wall models.......................................................................... 652 7.2.2.3. Vibroaoustic travelling waves in an “infinite” membrane ......................................................................................... 657 7.2.2.4. Sound transmission through an “infinite” membrane, or plate.............................................................................................. 661 7.2.2.5. Transmission through a finite plate.................................... 664 7.2.3. Radiation of water waves .................................................................. 668 7.2.3.1. Energy considerations........................................................ 668 7.2.3.2. Boundary value problem.................................................... 670 7.3. Dissipation induced by viscosity of the fluid................................................. 673 7.3.1. Viscous shear waves ......................................................................... 673 7.3.2. Fluid-structure coupling, incompressible case .................................. 677 7.3.2.1. Piston-fluid system ............................................................ 677 7.3.2.2. Flexible plates coupled by a liquid layer ........................... 682 7.3.2.3. Rigid plate coupled to a thin liquid layer........................... 687 7.3.2.4. Cylindrical annular gap...................................................... 690 7.3.2.5. Application to fluid induced damping of multisupported tubes......................................................................... 695 7.4. Dissipation in acoustic waves........................................................................ 697 7.4.1. Viscous dissipation ........................................................................... 697 7.4.1.1. Plane unconfined waves .................................................... 697 7.4.1.2. Importance of fluid confinement in viscous dissipation .... 699 7.4.1.3. Plane waves confined in a circular cylindrical tube........... 699 7.4.2. Miscellaneous dissipative mechanisms in acoustic waves................ 701 7.4.2.1. Heat conduction and thermoacoustic coupled waves ........ 702 7.4.2.2. Relaxation mechanisms ..................................................... 706

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Appendix A1. A few elements of thermodynamics ............................................ 708 A1. Thermodynamic refresher.............................................................................. 708 A1.1. Law of energy conservation .............................................................. 708 A1.2. Compressibility and thermal expansion coefficients......................... 710 A1.3. Second law: entropy.......................................................................... 710 A1.4. Maxwell relations.............................................................................. 711 A1.5. Thermodynamic relations particularized to perfect gases ................. 713 A1.6. Heat transfer and energy losses......................................................... 715 Appendix A2. Mechanical properties of common materials............................. 718 A2.1. Phase diagram............................................................................................. 718 A2.2. Gas properties ............................................................................................. 719 A2.3. Liquid properties......................................................................................... 720 A2.4. Solid properties........................................................................................... 723 Appendix A3. The Green identity ....................................................................... 724 Appendix A4. Bessel functions ............................................................................ 726 A4.1. Definition.................................................................................................... 726 A4.2. Bessel functions of the first kind ................................................................ 726 A4.3. Bessel functions of the second kind............................................................ 727 A4.4. Recurrence relations ................................................................................... 728 A4.5. Remarkable integrals .................................................................................. 729 A4.6. Lommel integrals ........................................................................................ 729 A4.7. Hankel functions......................................................................................... 729 A4.8. Asymptotic forms for large values of the argument.................................... 729 A4.9. Modified Bessel functions of the first and second kinds ............................ 730 Appendix A5. Spherical functions....................................................................... 732 A5.1. Legendre functions and polynomials .......................................................... 732 A5.2. Recurrence and orthogonality relations for Legendre polynomials ............ 735 A5.3. Spherical Bessel functions .......................................................................... 735 A5.4. Recurrence relations for spherical Bessel functions ................................... 736 A5.5. Spherical Hankel functions......................................................................... 737 Appendix A6. Specific impedances of several substances ................................. 739 References ............................................................................................................. 741 Index ...................................................................................................................... 748


Preface

In mechanical engineering, the needs for design analyses increase and diversify very fast. Our capacity for industrial renewal means we must face profound issues concerning efficiency, safety, reliability and life of mechanical components. At the same time, powerful software systems are now available to the designer for tackling incredibly complex problems using computers. As a consequence, computational mechanics is now a central tool for the practising engineer and is used at every step of the designing process. However, it cannot be emphasized enough that to make a proper use of the possibilities offered by computational mechanics, it is of crucial importance to gain first a thorough background in theoretical mechanics. As the computational process by itself has become largely an automatic task, the engineer, or scientist, must concentrate primarily in producing a tractable model of the physical problem to be analysed. The use of any software system either in a University laboratory, or in a Research department of an industrial company, requires that meaningful results be produced. This is only the case if sufficient effort was devoted to build an appropriate model, based on a sound theoretical analysis of the problem at hand. This often proves to be an intellectually demanding task, in which theoretical and pragmatic knowledge must be skilfully interwoven. To be successful in modelling, it is essential to resort to physical reasoning, in close relationship with the information of practical relevance. This series of four volumes is written as a self-contained textbook for engineering and physical science students who are studying structural mechanics and fluid-structure coupled systems at a graduate level. It should also appeal to engineers and researchers in applied mechanics. The four volumes, already available in French, deal respectively with Discrete Systems, Basic Structural Elements (beams, plates and shells), Fluid-Structure Interaction in the absence of permanent flow, and finally, Flow-Induced Vibrations. The purpose of the series is to equip the reader with a good understanding of a large variety of mechanical systems, based on a unifying theoretical framework. As the subject is obviously too vast to cover in an exhaustive way, presentation is deliberately restricted to those fundamental physical aspects and to the basic mathematical methods which constitute the backbone of any

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large software system currently used in mechanical engineering. Based on the experience gained as a research engineer in nuclear engineering at the French Atomic Commission, and on course notes offered to 2nd and 3rd year engineer students from ECOLE NATIONALE SUPERIEURE DES TECHNIQUES AVANCEES, Paris and to graduate students of Paris VI University, the style of presentation is to convey the main physical ideas and mathematical tools, in a progressive and comprehensible manner. The necessary mathematics is treated as an invaluable tool, but not as an end in itself. Considerable effort has been devoted to include a large number of worked exercises, especially selected for their relative simplicity and practical interest. They are discussed in some depth as enlightening illustrations of the basic ideas and concepts conveyed in the book. In this way, the text incorporates in a self-contained manner, introductory material on the mathematical theory, which can be understood even by students without in-depth mathematical training. Furthermore, many of the worked exercises are well suited for numerical simulations by using software like MATLAB, which was utilised by the author for the numerous calculations and figures incorporated in the text. Such exercises provide an invaluable training to familiarize the reader with the task of modelling a physical problem and of interpreting the results of numerical simulations. Finally, though not exhaustive the references included in the book are believed to be sufficient for directing the reader toward the more specialized and advanced literature concerning the specific subjects introduced in the book. To complete this work I largely benefited from the input and help of many people. Unfortunately, it is impossible to properly acknowledge here all of them individually. However, I whish to express my gratitude to Alain Hoffmann head of the Department of Mechanics and Technology at the Centre of Nuclear Studies of Saclay and to Pierre Sintes, Director of ENSTA who provided me with the opportunity to be Professor at ENSTA. A special word of thanks goes to my colleagues at ENSTA and at Saclay – Ziad Moumni, Laurent Rota, Emanuel de Langre, Ianis Politopoulos and Alain Millard – who assisted me very efficiently in teaching mechanics to the ENSTA students and who contributed significantly to the present book by pertinent suggestions and long discussions. Acknowledgments also go to the students themselves whose comments were also very stimulating and useful. I am also especially grateful to Professor Michael Païdoussis from McGill University Montreal, who encouraged me to produce an English edition of my book, which I found a quite challenging task afterwards! Finally, without the loving support and constant encouragement by my wife Françoise this book would not have materialized. François Axisa August, 2003

Introduction

Solid structures are generally in contact with at least one fluid. Therefore, the motion of the fluid and that of the solid are not independent from each other but constrained by a few kinematical and dynamical conditions which model the contact. As a corollary, the fluid and the structure, considered as a whole, behave as a dynamically coupled system. Going a step further, the motion can be split into a fluctuating and a permanent component. Such a distinction is extremely useful, conceptually at least, as it has profound implications concerning the physical behaviour and the mathematical modelling of the coupled system. As will be described in depth in the present volume, when there is no permanent motion, the fluid-structure coupled system is always dynamically stable. This is no more the case if a permanent flow exists and various dynamical instabilities can occur which have disastrous consequences on the mechanical integrity of the vibrating structures. To emphasize such a distinction which is of major concern to the engineer and also to the analyst, in this book by fluid-structure interaction we mean the dynamical coupling between a solid and a fluid in the absence of any permanent flow, whereas problems involving a permanent flow about a vibrating structure are referred to as flow-induced vibration problems. As will be described in volume 4 of this series, when dealing with a flow-induced vibration problem, it is appropriate to consider first the related fluid-structure interaction problem, by setting the permanent flow velocity field to zero. The dynamical behaviour thus obtained serves as a state of reference to investigate the coupling forces and dynamical response related to the interaction between the permanent flow and the vibrating structures. Hence, in accordance with the title, the present volume deals exclusively with modelling and analysis of the coupled oscillations of a fluid and a solid which occur about a state of equilibrium assumed to be stable and static. The subject is restricted essentially to the linear domain. It may be useful to emphasize that such problems are much more amenable to mathematical modelling and analysis than the flowinduced vibration problems. One basic reason is that fluid motions restricted to

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Introduction

oscillations or transients of small amplitude can be described as laminar flows, contrasting with most of the steady flows met in engineering which are turbulent. Practical relevance of fluid-structure interaction to engineering is nowadays asserted by a host of problems which are currently addressed to design structural components against excessive vibrations and noise in most industrial fields. Furthermore, the authors hope that the reader will be soon convinced that fluid-structure interaction problems present many fascinating and challenging aspects which make the study very appealing even if restricted to the linear domain. Furthermore, a few nonlinear problems will also be worked out, especially selected for their physical interest and relative easiness in analytical or numerical solution. This volume comprises seven chapters which describe distinct physical mechanisms leading to fluid oscillations eventually coupled to structural vibrations. Chapter 1 reviews the fundamental concepts and results of fluid mechanics used as a necessary background for the rest of the book. Fluid dynamics is then particularized to the case of small motions of a Newtonian fluid about a quiescent state. In this way, the physical properties of the fluid which control the fluid oscillations can be pointed out and described by a few steady quantities used to define the static state of reference of the fluid. Each quantity can be related to a distinct physical mechanism. Considering the particular case of harmonic oscillations, relative importance of the distinct mechanisms in a given system and a given frequency range can be measured by using a few dimensionless numbers. Such a preliminary study of fluid oscillations is used as a guideline to organize logically the content of the following chapters. Chapter 2 describes the fluid inertia effects, which are naturally related to the mean value of the fluid density. Fluid inertia affects the frequencies and the shapes of the vibration modes of the structures. Changes can be very significant, depending not only on fluid density but also on a few other features of the coupled system. It will be shown that inertia effects can be accounted for without adding any new degree of freedom to the coupled system. If a discretized model is used, they are entirely described by a so-called added-mass matrix which operates on the degrees of freedom of the structure. Chapter 3 presents the effects induced by gravity and surface tension at a free surface separating a liquid from a gas. Gravity and surface tension are found to provide the oscillating surface with restoring stiffness forces; in other terms they add some amount of potential energy to the vibrating system. As a consequence, fluid oscillations develop as surface waves. Presentation focuses on gravity waves which are of particular importance in naval and ocean engineering. The travelling waves, then the standing waves are described and finally their interaction with floating and grounded bodies is highlighted based on a few analytical examples. Chapters 4 and 5 are devoted to the acoustic waves which are related to the fluid elasticity. For the essential, this vast subject is restricted here to the aspects of relevance in the context of fluid-structure interaction, which concern essentially the

Introduction

xix

range of large wavelengths. Furthermore, at this step of presentation, dissipation is neglected. Chapter 4 deals with the one-dimensional case of plane waves in pipes. Special attention is paid to the transfer matrix method because of its intrinsic interest and because it stands as an efficient numerical tool to solve linear problems in complicated pipe networks. In Chapter 5, presentation is extended to 2D and 3D acoustics, considering successively standing and guided waves in acoustic enclosures and waveguides. Then the forced waves travelling in an unbounded space are addressed in some depth including the Kirchhoff-Helmholtz integrals which are the cornerstone of the boundary element methods used as an alternative to the finite element method for numerical solution of problems in unbounded space. Chapter 6 is concerned with the coupling between the acoustic waves and the structural vibrations. Special attention is paid to the case of low frequency vibrations in piping systems, for mathematical convenience in dealing with one-dimensional problems and also on account of the practical importance of the problem in many industrial plants which use extended pipe networks to convey various fluids. As will be shown, distinct formulations of the coupled vibroacoustic problem are possible depending on the variables used to describe the fluid. Semi-analytical solutions can be worked out by using the modal synthesis method, provided the modal density is not too large, which means in practice that the analysis is drastically restricted to the low-frequency range as soon as 2D and even more 3D problems are treated. Numerical simulations using the finite element method are particularly useful to deal with complicated structures coupled to a finite volume of fluid. General principles of the method have been described in Volume 2 in the context of structural mechanics. Hence, it is sufficient here to focus on the variational formulation of the fluid and the fluid-structure coupling terms, which are then discretized to obtained the finite element model of the fluid-structure system. To conclude this volume, dissipation mechanisms induced by the fluid oscillations are addressed in Chapter 7. In most vibration analyses, damping is an ingredient of paramount importance which, unfortunately, remains poorly amenable to prediction based on realistic physical models. The viscous damping model broadly used in design engineering to deal with lightly damped systems is first revisited in terms of dissipative waves and complex vibration modes. Two physical mechanisms of fluid dissipation are then described in depth, namely radiation damping and fluid viscosity. Finally other dissipative phenomena as heat conduction losses and relaxation mechanisms are only briefly evoked, since thorough descriptions can be found in several excellent textbooks in Acoustics. The content of the English version of the present volume has been considerably enlarged in comparison with the first Edition in French. Complements concern the description of the physical mechanisms, including new worked out illustrative examples, as well as the mathematical formalism and numerical techniques. Nevertheless, despite of the size, the present volume remains largely introductory in nature and by no way exhaustive compared with the present state of the art in the

xx

Introduction

field. Finally, the authors are especially grateful to Jean-François Sigrist for a thorough reading of the manuscript and interesting comments on the scientific and pedagogical content. A special word of thanks goes again to Philip Kogan, for checking and rechecking every part of the manuscript to improve the English and the editorial quality of the book. As in the preceding volumes of this series, any remaining errors and inaccuracies are purely the author’s own. François Axisa and José Antunes October 2006

Chapter 1

Introduction to fluid-structure coupling

This chapter is intended both as a qualitative preview of the various physical aspects of fluid-structure interaction and as a review of the basic equations which govern fluid dynamics. As the reader will see, fluid-structure interaction presents several intriguing and even fascinating aspects, which can be brought in evidence based on a few “simple” experiments. They are termed “simple” because they do not require any sophisticated test rig or instrumentation. However they call for a good sense of observation and deduction. Actually, fluid motion induced by a vibrating structure results from various distinct coupling mechanisms operating together, but with a relative importance which can vary enormously from one case to the other. Accordingly, it is extremely useful to start by identifying and modelling the individual mechanisms of interest. Concerning the formulation of the fluid dynamics, the purpose is restricted to introduce in a logical and synthetic way the formulation of the Navier-Stokes equations which govern the motion of a Newtonian fluid. Then, they are simplified to describe the linear oscillations of the fluid about a state of static and stable equilibrium (still, or stagnant, fluid) and the effects of different fluid boundary conditions. As an especially important result arising from such a theoretical analysis, a few dimensionless numbers are defined to assess the relative importance of each coupling mechanism entering into the fluidstructure interaction process, based on a few physical quantities of the fluidstructure coupled system.

2

Fluid-structure interaction

1.1. A short outline of fluid-structure coupled systems 1.1.1

Basic mechanism of fluid-structure dynamical coupling

Figure 1.1. Solid immersed in a fluid: (a) static case, (b) dynamical case

Let us consider a solid body immersed in a still fluid, as sketched in Figure 1.1. In this book, we are mainly interested in analysing the small vibrations of the solid, taking into account the physical mechanisms induced by the fluid. As in any vibration problem, it is first necessary to specify the state of equilibrium about which the fluid-structure system vibrates. In statics, the equilibrium of the solid is generally dependent on the static pressure field P0 ( r ) , which loads the solid at the wetted walls. It is of interest to notice that this static problem is not coupled, because P0 ( r ) is the solution of a hydrostatic problem which can be solved independently from the equilibrium configuration of the solid. In contrast with the static case, the dynamical problem is found to be coupled. Fluid-structure coupling can be understood based on the following mechanism of dynamical interaction: 1. Motion of the solid (fluctuating displacement field X s ( r ; t ) ) induces some motion within the fluid (fluctuating displacement field X f ( r ; t ) ), which is assumed to remain in contact with the solid without penetrating it. 2. As the fluid moves, fluctuating stresses σ f ( r ; t ) are generated (in particular a fluctuating pressure field p ( r ; t ) ), which load the solid by imposing fluctuating forces at the interface. As a consequence, the motion of the solid is modified. Of course, such a feedback mechanism can be reversed, starting from the motion of the fluid instead of that of the solid. Fluid-structure coupling can be modelled analytically based on the vibration equations of the solid and of the fluid, complemented with suitable coupling


3

conditions at the fluid-structure interface, that is at wall (W ) wetted by the fluid. They are given by the two following conditions: 1. On (W ) , the fluid and the solid have the same motion, because the fluid adheres to the wall. 2. The fluid and structural stresses exerted on (W ) are exactly balanced, because (W ) must be in local dynamical equilibrium. Depending whether the solid is totally immersed in the fluid, or not, (W ) is

defined as the whole boundary of the solid (Ss

) or as a part of it.

To conclude this subsection it is recalled that the dynamic equations of deformable solids are formulated within the framework of the theory of continuum mechanics by using a Lagrangian viewpoint. Hence, the physical quantities used to describe the motion are related to the material points, also called particles, of the continuous medium. In its initial (non deformed) configuration the solid occupies the volume (Vs ) bounded by the closed surface (Ss ) . The position vector r of a particle in the initial configuration and time t are used as independent variables. For instance X s ( r ; t ) denotes the vector field of displacement of a particle located initially at r . However, so long as the analysis is restricted to small elastic vibrations, it can be made based entirely on the initial configuration and the motion of solids can be suitably described using the structural elements models already established in [AXI 04,05]. The dynamic equations of fluids are also formulated within the framework of the theory of continuum mechanics. However, they are derived by using the Eulerian viewpoint according to which fluid properties such as density, pressure, velocity etc. are defined as fluctuating fields referenced to the space and not to the fluid particles. This is the approach classically followed in fluid dynamics. A first reason is that except for velocity which can be measured by using tracers which follow the flow to measure most of the fluid properties, for instance pressure, density and temperature, it would be very difficult to devise moving probes to follow fluid particles. Incidentally, the reverse is true in the case of solids, as probes like accelerometers or strain gages are fixed to the moving body. Furthermore, the mathematical description of fluid motion by using moving coordinates is complicated and used almost exclusively for implementing numerical techniques in finite element, or finite volume, computer codes. These techniques, known as Arbitrary Lagrangian Eulerian methods (in short ALE), are used to deal with problems involving large fluid and structural motions (see for instance [WAR 80], [SAR 98], [SOU 00], [CHU 02]). Clearly, such topics are far beyond the scope of the present book and will not be discussed further.

4


1.1.2

A few elementary experiments

Due to its mechanical properties, a fluid can modify the vibration of a structure through distinct physical mechanisms, whose relative importance may vary enormously from one case to the other. Before embarking on the task of modelling these mechanisms, it is of interest to get a qualitative preview based on a few experiments which can be performed rather easily by the layman.

Figure 1.2. Forced vibrations of a structure (your hand) immersed in a fluid (water)

A first experiment consists of waving nearly harmonically his/her hand in deep water, as sketched in Figure 1.2. It is clearly felt that the fluid resists the motion, though not preventing it. Further, it may be noticed that the resistive force increases in magnitude with the frequency and the amplitude of the oscillation. One major mechanism for this is related to the kinetic energy of the fluid set in motion by the solid. Adopting the hand motion as a scaling factor for the fluid motion, the kinetic energy of the fluid can be written as: Eκ( ) = f

1 ω2 M a X s2 ( t ) ∝ M a X 02 2 2

M a is known as the added mass coefficient which can be used to characterize the inertia force exerted by the fluid on the vibrating structure. M a is obviously

proportional to the fluid density ρ f . However, going a step further, it is also observed that the real motion of the fluid is much more complicated than that of the hand. Generally, it depends on the geometry and on the direction of the motion of the solid. This can be confirmed indirectly in the present experiment by observing that the fluid force varies to a large extent when the tilt angle of the hand with respect to the direction of motion is varied, or if the fingers are spread out. So, it can be expected that the calculation of M a is not a trivial task, except for a few


5

particularly simple geometries, scarcely met in practice. Historically, the first studies on fluid added mass was initiated by Du Buat (1786) who used it as a corrective term to determine accurately the period of pendulums, as reported in [STO 51], see also [NEL 86]. Inertial effects in dense fluid will be analysed in Chapter 2. To conclude on the hand waving experiment, it is also necessary to stress that the fluid force exerted on the hand is certainly not purely inertial in nature. It comprises also other components, in particular dissipative forces related to the fluid viscosity. However, to highlight viscous forces we prefer to consider the oscillations of a pendulum, as described later.

Figure 1.3. Surface waves triggered by a solid impacting the free surface of a liquid

The second experiment is done repeatedly by most human beings since childhood because the result is beautiful and intriguing. It simply consists of observing the surface waves triggered by throwing a pebble in still water, a pond for instance. At least three conclusions can be qualitatively drawn from the resulting wave pattern, which is idealized in Figure 1.3 based on a linear mathematical model of the impact by a spherical solid. First, the very existence of surface waves f indicates that some fluctuating potential energy Ep( ) is involved in the mechanism. Second, the free surface oscillates in the vertical direction whereas the waves progress in any radial direction at a speed which depends on the wavelength. Thus surface waves are recognized as being transverse and dispersive. Going a step further, one can observe that, depending on the size of the impacting sphere, the wave dispersive pattern differs strikingly. If the sphere diameter D is larger than about 2 cm, the wave speed increases with the wavelength, whereas the opposite occurs if D is less than a few millimetres. This indicates that the nature of the potential energy differs from one case to the other. Surface waves will be analysed in Chapter 3. It will be shown that in the first case, the potential energy is mainly related to the vertical component Z of the displacement of the fluid particles which oscillate in the earth’s gravity field of magnitude g. As will be proved in

6


subsection 1.3.2.2, the fluctuating gravity potential per unit area of a free surface is f Ep( ) = ρ f gZ 2 / 2 . Surface waves dominated by this mechanism are termed gravity waves. On the other hand, the capillary potential will be shown to be proportional to the surface tension coefficient σ f and to the square of the slope of the deformed free surface. Hence, as the length scale of the oscillations is shortened, the surface tension effect progressively prevails over the gravity effect. Another aspect of wave propagation which can be of importance in fluid-structure coupled problems is the radiation damping which can be conveniently highlighted by addressing the case of a floating object, the float of a fishing rod for instance.

Figure 1.4. Surface waves induced by an oscillating float

Figure 1.4 shows the progressive waves triggered by letting the float oscillate vertically. In case (a) we use the fishing rod to prescribe a nearly harmonic oscillation to the float. Accordingly, a nearly monochromatic surface wave is excited which travels in any radial direction. As a consequence, some mechanical energy is continuously radiated away by the waves. Provided the water extent is practically infinite, or limited by dissipative boundaries, the radiated energy is never returned to the vibrating body. The practical importance of radiation dissipation in the case of surface waves can be shown by letting the float oscillate freely after an initial impulse or vertical displacement. The existence of an oscillation indicates that the fluid provides some stiffness to the floating solid. Buoyancy stiffness is a mere consequence of the Archimedes force, which varies as the floating line of the solid is changed from its static level, in such a way that the stiffness force tends to bring the float back to its static state of equilibrium. It is also observed that the oscillation is heavily damped. The wave pattern observed some time after the initial excitation has stopped is sketched in Figure 1.4 (b). An external zone of still water is observed which obviously corresponds to the distances which are not yet reached by the first wave triggered at time t = 0. Then going further along an inward radial direction, the amplitude of the wave crests and troughs are found to decrease progressively,


7

being soon indiscernible. This, just because the wave amplitude is related to the amplitude of the float oscillation; so as time elapses, they become smaller and smaller. Radiation damping will be analysed in Chapter 7.

Figure 1.5. Surface standing waves or sloshing modes in a water tank shaken harmonically at an adjustable frequency

A third experiment concerning gravity waves consists of letting oscillate a halffilled water tank horizontally, as sketched in Figure 1.5. If the frequency is suitably tuned, the free surface is observed to oscillate vertically in a resonant manner. At resonance, an excitation of very small amplitude is sufficient to induce water sloshing with large amplitude, leading eventually to water spilling out of the vessel, whereas outside the resonant domain, amplitude of the oscillations is drastically reduced, usually by at least two orders of magnitude. By sweeping the excitation frequency progressively through a fairly large range of values, the number of such resonant fluid responses is found to increase steadily with the frequency range explored. As in the case of the natural modes of vibration of a solid structure, sloshing modes arise as standing gravity waves due to the confinement of the liquid by the reflecting walls of the vessel. Furthermore, if the walls are flexible, the coupling of the sloshing modes with the structural modes of vibration can occur, leading to new natural modes of vibration of the fluid-structure coupled system which are marked by structural and fluid oscillations occurring at the same natural frequency. Sloshing modes and coupling with structural modes of vibration will be analysed in Chapter 3. Energy dissipation due to fluid viscous friction can be conveniently brought in evidence by oscillating freely a pendulum in various fluids, for instance air and water. Historically, the pendulum was used for that purpose from the first half of the nineteen century, in particular by Bessel (1828), Poisson (1831) Bailey (1832) and G.G. Stokes. The outstanding studies by Stokes [STO 51] led the author to define the “index of fluid friction” i.e. the dynamic viscosity coefficient as it was termed later. Furthermore, it is also of interest to use aerodynamically profiled and bluff body shapes to build the pendulum. The latter is released at an initial angle θ 0 from

8


the vertical position, as sketched in Figure 1.6. Provided θ 0 is sufficiently small, a damped harmonic oscillation is observed at the natural frequency of the pendulum which fades out progressively with time. If the test is carried out first in air, or even better in vacuum, and then in water, the following points are noticed:

Figure 1.6. Free oscillations of a pendulum immersed in a viscous fluid

1. The natural frequency of the oscillations is lower in liquid than in air, or vacuum ( ω L < ω 0 ). Because of the large increase of fluid density when passing from air to water, inertia of the oscillating liquid increases the equivalent mass M e of the pendulum, whereas the effective weight of the pendulum is lowered, in agreement with Archimedes’s theorem. Therefore, the stiffness of the pendulum diminishes due to the buoyancy effect. As a limiting case, if the density of the solid becomes less than that of the liquid, the lower position of static equilibrium θ s = 0 becomes unstable and the pendulum oscillates about the upper position θ s = π. 2. In air, small damping ratios within the range 10−4 ≤ ς 0 ≤ 10−3 can be easily achieved. When the experiment is carried out in water, damping is increased, as expected since the dynamic viscosity of water is much larger than that of air. However, things are not as simple as that, because the geometry of the immersed solid and the magnitude of vibrations are also found to be of paramount importance. In the case of a body profiled in such a way that practically no flow separation occurs, damping ratios within the range 10−3 ≤ ς L ≤ 10−2 can be observed, which increase progressively with the fluid viscosity as conveniently studied by using aqueous glycerine solutions of different concentrations. In contrast, if a bluff body is used, a cube or a sphere for instance, together with a large value of θ 0 (about 20° for instance) the first oscillations are found to be


9

heavily damped, whereas damping is substantially less as soon as the crest amplitude θ m ( t ) of the oscillations become less than a certain value. Dissipative effects due to fluid friction will be further discussed in Chapter 7.

Figure 1.7. Percussive musical instrument: drum

Fluid elasticity (or compressibility) gives rise to dilatational waves, which are quite similar to the dilatational waves observed in solids (see for instance [AXI 05]). Such waves are also known as sound waves, especially if their frequency lies in the audio-frequency range, which extends roughly from about 20 to 20000 Hz. Sound waves are often produced by letting vibrate a structural element immersed in a fluid. This is the case of a large host of many musical instruments. The example of a drum beaten by a stick is sketched in Figure 1.7. A sound at the excited frequencies can be heard at any place within the fluid, external, or internal. Acoustical waves travel through the air outside the drum. The air enclosed within the drum experiences stationary waves, the so called acoustical resonances, which in fact are often perceptibly modified by the coupling mechanism with the natural modes of the tensioned membrane. Acoustical plane waves will be studied in Chapter 4 and the three-dimensional case will be studied in Chapter 5. Finally, Chapter 6 will deal with the interaction between flexible solids and compressible fluids, which gives rise to the vibroacoustic modes marked by both structural and fluid oscillations. Incidentally, existence of such coupled modes is of paramount importance to govern the resonance frequencies and then the pitch of kettle drums as shall be demonstrated in Chapter 6 subsection 6.4.2. To conclude this preliminary survey, it is stressed that, in the experiments briefly described just above, the fluid is set into motion by the solid vibration solely. As a consequence, the fluid-structure coupled system vibrates about a state of stable equilibrium, in which both the fluid and the structure are motionless. When the oscillations occur in a flowing fluid, other effects than those mentioned just above, take place. As could be expected from the difficulties encountered in theoretical fluid dynamics, modelling of fluid-structure interaction problems is a far easier task in the former case than in the second one, especially when the flow is turbulent.

10


Hence, it is found appropriate to make a clear distinction between two broad classes of fluid-structure interaction problems, namely that of the fluid-structure coupling about a stagnant state, and that of the flow induced vibrations. Fluid-structure coupling is the object of the present Volume and flow induced vibrations will be that of Volume 4. 1.2. Dynamic equations of fluid-structure coupled systems 1.2.1

Elastic vibrations of solid structures

We consider a solid modelled as a continuum medium which occupies a finite volume (Vs ) bounded by a closed surface (Ss ) . As explained for instance in [AXI 05], the motion of the solid is governed by the local equations of dynamical equilibrium, also called momentum equations, written here as: ρ s X s − div σ s = f s( e ) ( r ; t ) ; ∀ r ∈ (Vs ) σ s ( r ).n ( r ) − K s( S ) ⎡⎣ X s ⎤⎦ = ts( e ) ( r ; t ) ; ∀ r ∈ (Ss ) [1.1] (e) (e) 1 X s ( r ; t ) = X s ( r ; t ) ; ts ( r ; t ) ≡ 0 ∀ r ∈ Ss( )

( )

In this system, ρ s is the density (mass per unit volume) and σ s is the Cauchy stress tensor of the solid material. The body is subjected to an external loading which is time dependent (fluctuating loads). It may comprise: e 1. A body force field f s( ) ( r ; t ) (force per unit volume) assumed to vanish at the boundary. e 2. A contact force field ts( ) ( r ; t ) (force per unit area) exerted on the boundary.

( )

3. A motion prescribed to a portion Ss( ) displacement field X s( e ) ( r ; t ) for instance. 1

of the boundary, as a given

Prescribed velocities or accelerations would be formulated in the same way as a e prescribed displacement field. Furthermore, it is necessary to assume that ts( ) ( r ; t )

( ) since it would be inconsistent to prescribe an external force and

vanishes on Ss( ) 1

a given motion to the same point. Finally, the body is assumed to be provided with elastic supports, described by some stiffness operator K s( S ) defined on (Ss ) . The unit vector n ( r ; t ) normal to (Ss ) is conventionally directed from the solid to the external medium. In the case of small displacements, any change between the initial and the actual configuration of the solid can be neglected. As the solid moves, its


11

initial and actual configuration at a later time t, are not the same. Nevertheless, so long as the theory is restricted to the linear domain, displacements and strains must be assumed to be so small that any change in the configuration of the solid can be safely discarded. On the other hand, in the case of linear elasticity, the Cauchy stress tensor of an isotropic material is written as: [1.2] σ s = λs Tr ε s I + 2 μ s ε s = λs div X s I + 2 Gε s

( )

(

)

where I stands for the identity tensor. The elastic coefficients are expressed either in terms of the elastic Lamé parameters λs and μ s (the shear modulus μ s is often denoted G in structural engineering), or in terms of Young’s modulus Es and Poisson’s ratio ν s . The relations between these parameters are: λs =

ν s Es Es ; μs = G = 2 (1 + ν s ) (1 + ν s )(1 − 2ν s )

[1.3]

The tensor ε s of small strains is defined as: T⎞ 1⎛ ε s = ⎜ grad X s + ⎛⎜ grad X s ⎞⎟ ⎟ 2⎝ ⎝ ⎠ ⎠

The upper script

T

( )

[1.4]

stands for a matrix transposition and the double bar over the

gradient operating on a vector is used to emphasize that it produces a second rank tensor. Similarly, an arrow over the gradient operating on a scalar marks that the result is a vector. Substituting [1.2] into [1.1], the equations of motion of a solid modelled as a 3D elastic medium, are formulated as: ρ s X s − ⎡⎣G Δ X s + (λs + G ) grad ⎡⎣ div X s ⎤⎦ ⎤⎦ = f s( e ) ( r ; t ) ∀ r ∈ (Vs ) T⎞ ⎛ λs div X s I .n + G ⎜ grad X s + ⎛⎜ grad X s ⎞⎟ ⎟ .n − K s ⎡⎣ X s ⎤⎦ = ts( e ) ( r ; t ) ∀ r ∈ (Ss ) ⎝ ⎠ [1.5] ⎝ ⎠ (e) (e) X s ( r ; t ) = X s ( r ; t ) ; ts ( r ; t ) ≡ 0 ∀ r ⊂ Ss(1) Vs ( t ) ≡ Vs ( 0 ) = Vs

;

Ss ( t ) ≡ Ss ( 0 ) = Ss

( )

The first line of this system stands for the vibration equation, the second line is the equilibrium balance at the boundary, maintained by elastic supports and loaded by an external contact fluctuating force. The third line gives the conditions of equilibrium on that part of the boundary which is loaded by a prescribed motion. The final line specifies that the configuration about which the solid vibrates is time independent. As emphasized in [AXI 05], the vibration equation is of the canonical form:

12


[1.6] M s ⎡⎢ X s ⎤⎥ + K s ⎡⎣ X s ⎤⎦ = f s( e ) ( r ; t ) ⎣ ⎦ M s ( r ) and K s ( r ) are the mass and stiffness operators of the solid. It is recalled that they are self-adjoint. In addition, M s ( r ) is positive definite and K s ( r ) is positive. The natural modes of vibration of the solid are determined by solving the homogeneous problem: K s ⎡⎣ X s ⎤⎦ − ω 2 M s ⎡⎣ X s ⎤⎦ = 0 ; ∀ r ∈ (Vs ) [1.7] σ s ( r ).n ( r ) − K s( S ) ⎡⎣ X s ⎤⎦ = 0 ; ∀ r ∈ (Ss )

Structural elements are modelled by using an equivalent 1D, or 2D-medium by formulating a set of simplifying assumptions concerning the deformations of the body, based on the geometry of the structure. The resulting equations of vibration are also of the canonical form [1.6]. 1.2.2

Dynamic equations of Newtonian fluids

1.2.2.1 Eulerian acceleration and material derivative As already indicated in subsection 1.1.1, fluid dynamics is generally formulated by using Eulerian fields. As an Eulerian field refers to fixed points of the geometrical space and not to the material points, the time rate of change of any physical quantity related to a specific infinitesimal part of the fluid (the so-called fluid particle) has to be formulated differently than in the case of a Lagrangian field, where the ordinary time partial derivative ∂ [ ] / ∂t is appropriate. In the case of an Eulerian field the latter has to be replaced by the substantial derivative operator, identified by the symbolic notation D [ ] / Dt , as devised by Stokes. For instance, the acceleration of a fluid particle is written as DV / Dt , where the fluid velocity is noted V instead of X f to alleviate the notation. To express the substantial derivative in terms of ordinary partial derivatives, we consider a particle located at r at time t and at r + δ r slightly later (time t + δ t ). Provided δ t is small enough a Taylor expansion of the Eulerian field of velocity up to the first order is sufficiently accurate, leading to: ⎞ ⎛∂V [1.8] + V.grad V ⎟ V(r + δ r;t + δ t ) = V(r;t) + δ t ⎜ ⎝ ∂t ⎠ r ,t Letting δ t tend to zero, the substantial derivative is expressed as:


⎞ DV ⎛ ∂ V =⎜ + V.grad V ⎟ Dt ⎝ ∂ t ⎠

13

[1.9]

The same reasoning applies to the time rate of change of any material quantity defined as an Eulerian field. For instance, the substantial derivative of a scalar S is found to be: DS ∂ S = + V.grad S Dt ∂ t

[1.10]

The first term in the right-hand side of [1.10] is called the local rate of change because it vanishes if S is constant. Of course, it can be identified with the rate of change of S, if described by using a Lagrangian field. The second term is called the convective rate of change. It means that the space variations of S are convected into the fixed point r by the flow velocity V . So it differs from zero, unless S is uniform along the direction of V . A slightly different way to interpret the substantial derivative is in terms of frame transformation. Let be ( Σ ) the inertial frame to which [1.10] is referred and ( Σ ′) the frame moving at velocity V with respect to ( Σ ) . In ( Σ ′) a particle located at r ′ at time t is still located at r ′ at time t + δ t . Therefore, ( Σ ′) is called the co-moving frame and the substantial derivative

defined in ( Σ ) is equal to the partial derivative with respect to t ′ = t in the comoving frame ( Σ ′) .

1.2.2.2 Mass conservation equation Assuming that during the motion there is no loss or gain of matter, the rate of change of the mass M f of fluid contained in a fixed volume (Vf ) (the so-called control volume), must be balanced by the mass flux Qf through the boundary (S f ) of the control volume. M f and Qf are given by: ⌠

M f = ⎮⎮

⌡(Vf

)

ρ f dV

;

⌠

Qf = ⎮⎮

⌡(S f )

⌠ ρ f V.n dS = ⎮⎮

⌡(Vf

div( ρ f V)dV

[1.11]

)

The unit vector n ( r ; t ) normal to (S f ) is conventionally directed from the fluid to the external medium. ρ f V .n is the mass flux per unit area, expressed in kgs-1m -2 .

Thus the mass balance is:

14 ⌠ ⎮ ⎮ ⎮ ⎮ ⌡(Vf


)

⎞ ⎛∂ Mf + div( ρ f V ) ⎟dV = 0 ⎜ ⎝ ∂t ⎠

[1.12]

As the control volume is arbitrary, the global balance [1.12] implies that the local balance also holds, hereafter referred to as the mass equation: Dρ f + div( ρ f V ) = + ρ f div V = 0 ∂t Dt

∂ ρf

[1.13]

where the second expression in [1.13] is derived from the first one by using the mathematical identity div( ρ f V ) = ρ f div V + V .grad ρ f in conjunction with [1.10]. If an external source of fluid characterized by the mass per unit volume m (f ) , e

expressed in kgm -3 , is injected at time t into the control volume, the mass balance becomes: ⌠ ⎮ ⎮ ⎮ ⎮ ⌡(Vf

⌠

)

⎮ ⎞ ⎛∂ Mf + div( ρ f V ) ⎟dV = ⎮⎮ ⎜ ⎝ ∂t ⎠ ⎮ ⎮

⌡(Vf

∂ m (f ) e

)

∂t

dV ⇔

∂ m (fe ) + div( ρ f V ) = ∂t ∂t

∂ ρf

[1.14]

On the other hand, it turns out that in many cases fluid compressibility can be neglected to a high degree of accuracy. According to the incompressible model, the volume and the density are assumed to remain constant. So, the mass equation [1.13] simplifies into: div V = div X f = 0 [1.15] which for centred oscillations is equivalent to the condition of incompressibility in statics div X f = 0 . 1.2.2.3 Momentum equation The momentum equation is of the same form as equation [1.1], provided the inertia force is expressed by using the Eulerian acceleration [1.9]. The result is the Navier equation: ρf

e DV − div σ f = f f( ) Dt

[1.16]

which governs the rate of change of the fluid momentum. Discussion of the boundary conditions is postponed to subsection 1.3.2. To describe the stresses


15

arising in a flowing fluid, it is found appropriate to start by introducing successively the concepts of pressure and viscosity. 1.2.2.4 Pressure and fluid elasticity According to experiment, in statics, a fluid can resist an external load through normal stresses only, which are the same in every direction, provided the fluid is isotropic. The so-called hydrostatic stresses are thus given by the isotropic diagonal tensor: ⎡P 0 σ f = −⎢0 P ⎢ ⎢⎣ 0 0

0⎤ 0⎥ ⎥ P ⎥⎦

[1.17]

P is the pressure, positive if the fluid is compressed. Therefore, contrasting with the case of solids, no static shear stresses arise in a fluid to resist an external shearing load. Using [1.17], the stress forces per unit volume are expressed as: div σ f = −grad P [1.18] Such a mechanical definition of pressure is appropriate in particular when the fluid is assumed to be incompressible. At this respect it is of interest to recall here the example already presented in [AXI 05] Chapter 1, of a water column enclosed in a e rigid tube and loaded by an axial contact force T ( ) through a waterproof piston, see Figure 1.8. We are interested in determining the pressure field in the fluid. Let us assume that the problem is one-dimensional, as reasonably expected and justified later in Chapters 2 and 4.

Figure 1.8. Column of liquid compressed in a rigid tube

Obviously, the condition of local and/or global static mechanical equilibrium leads immediately to a uniform pressure P = −T ( e ) / S f where S f is the tube crosssectional area (section normal to the piston axis). This result is clearly independent of the material law of the fluid. Going a step further, we want to define the pressure in a logical manner starting from the material behaviour of the fluid, here the law of incompressibility. The appropriate manner is to determine the pressure by using the

16


Lagrange multiplier associated with the holonomic condition ∂ X f / ∂ x = 0 . The variation of the constrained Lagrangian is: L

⌠ ∂ (δ X f ) dx + T ( e )δ X f ( L ) = 0 δ L' = Sf ⎮ Λ ∂x ⎮ ⌡0 After integrating by parts the above expression is transformed into: L

⌠ ∂Λ L e δ L ' = −S f ⎮ δ X f dx + ⎡⎣ Λ S f δ X f ⎤⎦ + T ( )δ X f ( L ) = 0 0 ⎮ ∂x ⌡0 The variation δ X f is arbitrary, but admissible. Accordingly, at the bottom of the fixed and rigid tube δ X ( 0 ) = 0 and the expected result is obtained as follows: ∂Λ ∂P = =0 ∂x ∂x

; Λ=

−T ( e ) =P Sf

In dynamics, as in statics, mechanical pressure can be defined in a more formal way as being equal to one third of the trace of the stress tensor: 1 P = − Tr ⎡⎢σ f ⎤⎥ 3 ⎣ ⎦

[1.19]

Fluid compressibility does not modify the pressure which balances the external load, but provides a mean to relate the pressure to the fluid strain through a strainstress relationship similar to that used in the case of an elastic solid. In this respect, the system depicted in Figure 1.9 can be viewed as the fluid counterpart of the bar used in a tensile test machine to determine the monoaxial strain-tress relationship of solids. Restraining the discussion to the case of linear elasticity, it is appropriate to consider small changes between two states of static equilibrium. Here, the thermodynamic aspect of the problem is discarded for a while, by assuming that temperature T0 is constant and equal to the room temperature. In state (1) the external load is T ( ) , the fluid column occupies the volume V0 = L0 S f , density is ρ0 e

and pressure P0 is equal to −T ( e ) / S f . In state ( 2 ) the force balance is: − S f ( P0 + δ P ) = T0( ) + δ T0( e

e)

[1.20]


17

Figure 1.9. Compressible fluid column compressed in a rigid tube

The column is compressed or expanded by the amount δ L = X , which is found to e be proportional to δ T ( ) , if the latter does not exceed a certain value, which may largely depend on the nature of the fluid (liquid or gas) and on the values P0 , T0 of the state of reference (1). So, the isothermal Young modulus of a fluid can be defined in the same way as in the case of a solid by Hooke’s law: δ P(

T0 )

= − E (f 0 ) T

δ L( 0 ) L0 T

where the upper script

[1.21]

( ) T0

is used here to specify that the transformation is

isothermal. The minus sign indicates that pressure is increased if the fluid column is contracted. On the other hand, as the fluid mass is constant, the change in density is also proportional to the axial deformation of the fluid column: ⎛ δL⎞ M f = nM = ρ 0V0 = ( ρ 0 + δρ )(V0 + δV ) = ( ρ 0 + δρ )V0 ⎜ 1 + ⎟ L0 ⎠ ⎝

[1.22]

M is the molecular mass and n the number of moles in the fluid column. By eliminating the axial deformation between equations [1.21] and [1.22], the isothermal Young modulus can also be expressed as: ⎛δ P ⎞ ⎛ ∂P ⎞ 1 T = ρ0 ⎜ = (T0 ) E (f 0 ) = ρ0 ⎜ ⎟ ⎟ ∂ δρ ρ ⎝ ⎠ (T0 ) ⎝ ⎠ (T0 ) κ f

κ (f 0 ) is the coefficient of isothermal compressibility of the fluid, defined as: T

[1.23]

18


κ (f 0 ) = − T

1 ⎛ ∂V ⎞ 1 ⎛ ∂ρ ⎞ = ⎜ ⎟ ⎜ ⎟ V0 ⎝ ∂P ⎠ (T0 ) ρ 0 ⎝ ∂P ⎠ (T0 )

[1.24]

Noticing that ∂P / ∂ρ has the dimension of a velocity squared, relation [1.24] may be rewritten as: Ef ρf

=

∂P = c 2f ∂ρ

Here, c f

[1.25]

denotes the speed of sound in the fluid without specifying the

thermodynamical conditions of the pressure and density changes, which is thus defined in the same way as in the case of the elastic dilatational waves in a solid with no Poisson effect (Poisson ratio ν s = 0 ). On the other hand, as in the case of a solid, the elastic potential density per unit of fluid volume is: 1 ee = σ f : ε f 2

[1.26]

ε f is the small strain tensor. Using the hydrostatic stress tensor [1.17], formula

[1.26] is written as: ee = −

1 p div X f 2

[1.27]

X f is the (small) displacement field of the fluid and p stands for the small change

in pressure. The one-dimensional elastic law [1.21] is extended to the 3D case as: p = − E f divX f = − ρ f c 2f divX f [1.28] Substituting [1.28] into (1.27] we arrive at the quadratic and positive form: ee =

1 ρ f c 2f div X f 2

(

)

2

=

p2 2 ρ f c 2f

[1.29]

1.2.2.5 Fluid elasticity and equation of state of a gas It is also useful to introduce the fluid properties based on a few thermodynamical considerations. The basic concepts and relations of thermodynamics needed in this book are briefly recalled in Appendix A1. From the viewpoint of thermodynamics, pressure arises as a quantity governed by an equation of state which relates pressure to two other thermodynamic independent variables, for instance density and temperature. This equation may be viewed as an elastic law, which generally is


19

nonlinear. A classical example of such a state equation is the perfect gas law, written here as: P R = T ρ M

[1.30]

R = 8.314 Joule/mole °K designates the universal gas constant and T is the absolute temperature in °K. However, in the absence of any additional hypothesis concerning the transformation, T is an independent variable and the connection between P and ρ can not be entirely specified. As two extreme cases, the transformation can be assumed to be either isothermal, or adiabatic. As shown in Appendix A1, formula [A1-43], the whole range of possibilities is conveniently modelled by using the so called polytropic law: −γ p

P1 ρ1

−γ

= P2 ρ 2 p = constant

[1.31]

where the subscripts (1) and (2) refer to two distinct states of the gas. The polytropic index γ p is equal to unity if the transformation is isothermal, and is equal to the ratio of the specific heats at constant pressure and at constant volume: γ p = γ = C p / CV , if the transformation is adiabatic. Linearizing the equation of state about the equilibrium state ρ0 , P0 , T0 , the small changes p and ρ are found to be related by a relation of the type: p = ρ c02

[1.32]

where the subscript

(0)

marks that the value of the sound speed refers to the static

state of equilibrium, i.e. the so-called quiescent or still fluid, see Figure 1.10. Application to the case of a perfect gas is straightforward. P0 and ρ0 are related to each other by the perfect gas law [1.30], and the small changes about these values are related by the polytropic law [1.31], the speed of sound is expressed as: c0 = γ p

γ pR P0 = T0 M ρ0

[1.33]

Young’s modulus is: E0 = γ p P0

[1.34]

In particular, at standard temperature θ 0 = 20°C and pressure conditions Pa 1bar (in short STP), the Young’s modulus of atmospheric air is comprised between 1 and 1.4 bar depending whether the transformation is isothermal, or adiabatic. Further discussion on the practical importance of heat transfer in acoustical waves is postponed to Chapter 4, concerning elasticity and to Chapter 7 concerning damping.

20


Figure 1.10. Equation of state of a fluid and sound speed

Contrasting with such a highly compressible behaviour, liquids are poorly compressible. Seawater for instance is found to have a density varying from about 1002 kg/m -3 at the ocean surface to about 1007 kg/m -3 at 10,000 meters depth ( ≈ 1 kbar ). Correlatively the coefficient of compressibility at 0°C and 1 bar is only 4.610−5 per bar, a value which varies very slightly with pressure and temperature. In accordance with [1.23], Young’s modulus of water is about 2109 Pa instead of 105 Pa in atmospheric air, which means that air at standard conditions is more compressible than water by a factor of about 20,000. To conclude this subsection it is important to emphasize that, in statics, the pressure derived from the mechanical equilibrium must be the same as the pressure derived from the thermodynamic equilibrium, because the system could not remain in a static state unless both mechanical and thermodynamic equilibrium conditions are fulfilled. A priori, thermodynamic and mechanical pressures are not necessarily the same in a moving fluid, because the time scale to reach a thermodynamic equilibrium can be larger than the time scale of motion, as briefly outlined in Chapter 7 in relation to relaxation mechanisms and sound absorption. 1.2.2.6 Cavitation of a liquid Experiment shows that pressure in a liquid cannot be diminished below a certain threshold value Pc , termed pressure of cavitation. If gas, in suspension or dissolved, is not carefully removed from the liquid, Pc is nearly equal to the saturating pressure of the liquid. When the threshold value is reached, if the fluid is further stretched, liquid vaporizes and pressure remains constant. In the presence of dissolved gas, as the pressure decreases bubbles are generated even before vaporization is initiated. The phenomenon presents a marked similarity with plasticity in ductile solids, as sketched in Figure 1.11. However, in contrast to plasticity, when the cavitating liquid is compressed again, the vapour bubbles, or pockets, collapse very quickly in


21

such a way that strong short-lived pressure transients are generated. In this respect, recompression of a cavitating liquid presents a marked similarity with impacts between stiff solids. Dynamics of cavitation germs and bubbles will be discussed in Chapter 3, and modelling of dynamical effects of cavitation in piping systems will be outlined in Chapter 6.

Figure 1.11. Pressure in a cavitating fluid

1.2.2.7 Viscous stresses Viscosity can be suitably introduced with the aid of a conceptual experiment in which a fluid layer of thickness h is bounded by two parallel plates, see Figure 1.12. The plates are moved in their own plane along the Ox direction, in such a way that no normal stress is induced in the fluid. In the absence of viscosity, the plates would slide freely and the fluid would remain at rest. However, as experiment shows, the fluid adheres to both walls in such a way that fluid velocities at z = 0 and z = h are equal to the velocity of the adjacent plate. Furthermore the whole layer is set into motion in the Ox direction. This clearly requires the presence of tangential (or shear) stresses σ zx acting in the Ox direction.

Figure 1.12. Velocity distribution in a fluid layer bounded by two parallel flat walls

22


By assuming a linear relationship between σ zx and the fluid velocity, one is led naturally to postulate the Newton law of fluid friction, written in the onedimensional case as: σ zx = + μ f

∂V ( z ) ∂z

[1.35]

The proportionality constant μ f > 0 stands for a fluid property called the coefficient of dynamic viscosity. This physical quantity is assumed to be positive and the positive sign in [1.35] indicates that the viscous force accelerates the fluid and opposes the plate motion, in agreement with the principle of action and reaction. Furthermore, the fact that the Newton law is proportional to the gradient of the fluid velocity and not the velocity itself, can be understood by considering the uniform motion induced by starting the two plates from rest to a steady motion in which they have the same constant speed. During the accelerated part of motion, the fluid is accelerated too, while during the steady part, both fluid and solid move at the same constant speed and no force develops in the system, in agreement with the Galilean principle of inertia. Considering two adjacent fluid layers flowing at distinct speeds, the friction law [1.35] implies that the fastest layer is decelerated whereas the slowest is accelerated, what requires mechanical energy. Thus the trend of viscous forces is to reduce the velocity gradients, at the cost of a loss of energy. In Figure 1.12, the upper plate moves at the constant velocity V and the lower plate is fixed. Provided V is sufficiently small, the velocity profile inside the fluid layer is linear v(z) = (z/h)V. This laminar flow regime (Couette’s flow) subsides beyond a certain value of V, the flow becoming three dimensional and irregular (turbulent). The Newton friction law [1.35] holds for laminar flows solely. To deal with three-dimensional laminar flows, it can be suitably extended into a tensor form by using the velocity gradient. Furthermore, if the fluid is isotropic it can be shown that the viscous stress tensor is necessarily of the symmetrical form (see for instance [JEF 63]): σ v = 2 μ f ε f + λ f Tr ⎡⎢ε f ⎤⎥ I ⎣ ⎦

[1.36]

where ε f is the rate of strains tensor, defined in a similar way as [1.4]: ⎞T ⎞ ⎛ 1⎛ ε f = ⎜ grad X f + ⎜ grad X f ⎟ ⎟ 2⎜ ⎝ ⎠ ⎟⎠ ⎝

[1.37]

As in the case of an isotropic and elastic solid, the strain-stress law depends on two material coefficients only, noted μ f and λ f . However, their physical meaning is completely distinct from that of the Lamé elastic parameters. They are called the first and second viscosity coefficients, respectively. It is noticed that


23

λ f dependency disappears if fluid compressibility is neglected. By adding the

viscous stress tensor to the hydrostatic pressure tensor we obtain: σ f = − PI + 2 μ f ε f + λ f Tr ⎡⎢ε f ⎤⎥ I ⎣ ⎦

[1.38]

The mechanical pressure as defined by [1.19] is found to be: 1 2 ⎞ ⎛ P = − Tr ⎢⎡σ f ⎥⎤ = P − ⎜ λ f + μ f ⎟ div X f 3 ⎣ ⎦ 3 ⎠ ⎝

( )

[1.39]

If compressibility is discarded, mechanical and thermodynamic pressure are the same and equal to the hydrostatic pressure. Going a step further, Stokes made the assumption that this equality still holds in compressible flowing fluid, which implies that: 2 λf + μ f = 0 3

[1.40]

With the aid of [1.40], the Stokes stress tensor is finally written as: 1 ⎛ ⎞ σ f = − PI + 2 μ f ⎜ ε f − Tr ⎡⎢ε f ⎤⎥ I ⎟ ⎣ ⎦ 3 ⎝ ⎠

[1.41]

where no distinction is made between mechanical and thermodynamic pressure. Most of the common fluids, termed isotropic Newtonian fluids, are satisfactorily described by the tensor [1.41], except if relaxation mechanisms are to be accounted for, as outlined in Chapter 7. Viscosity of such fluids is entirely characterized by the dynamic coefficient of viscosity μ f . It is also useful to use the kinematic coefficient of viscosity defined as follows: νf =

μf ρf

[1.42]

The values of μ f and ν f are found to be very sensitive to temperature and to the nature of the fluid (see tables of Appendix A2). 1.2.2.8 Navier-Stokes equations The momentum equation of an isotropic Newtonian fluid is obtained by substituting the Stokes tensor into equation [1.16] as: ⎞ e ∂V ⎛ 1 ρf + ρ f V .gradV + gradP − μ f ⎜ ΔV + grad ⎡⎣ divV ⎤⎦ ⎟ = f f( ) [1.43] ∂t 3 ⎝ ⎠

24


where Δ = div ⎡⎣ grad [ ]⎤⎦ denotes the Laplacian operator. The so-called Navier-Stokes equations are formed by gathering together the mass equation [1.14] and the momentum equation [1.43]. If viscosity is neglected, which corresponds to the so-called inviscid fluid model, they reduce to the Euler equations. On the other hand, the inertial terms can be rearranged to obtain two other equivalent forms of interest. The first one reads as: ∂ ρ fV ∂V [1.44] ρf + ρ f V .gradV = + div( ρ f VV ) ∂t ∂t The right-hand side of [1.44] can be established by using the two following identities: ∂ V ∂ ρ fV ∂ ρ f ∂ ρ fV [1.45] ρf = −V = + Vdiv( ρ f V ) ∂t ∂t ∂t ∂t

div( ρ f VV ) = ρ f V .grad V + V div( ρ f V )

[1.46]

In Cartesian coordinates, the momentum flux tensor P = ρ f VV is written in matrix

form as: ⎡VxVx VxVy VxVz ⎤ ⎢ ⎥ ρ f VV = ρ f ⎢V yVx V yV y V yVz ⎥ ⎢VzVx VzVy VzVz ⎥ ⎣ ⎦

[1.47]

The flux of momentum per unit area, through a surface of unit normal vector n is:

P = P.n

[1.48]

By using the following vector identity: 2 B.gradA = grad( B. A) + curl( A × B) + B div A − A div B − B × curl A − A × curl B the even more interesting form can be obtained: ⎞ ⎛ ∂ V 1 2 ∂V ρf + ρ f V .gradV = ρ f ⎜ + gradV + (curlV ) × V ⎟ ∂t ⎝ ∂t 2 ⎠

[1.49]

The right-hand side of [1.49] allows one to split the nonlinear inertia term into a component deriving from a potential and a rotational component. The latter is often expressed in terms of the vorticity vector Ω , defined as:


25

1 Ω = curl V [1.50] 2 As Ω dt represents an infinitesimal rotation of the continuum (see for instance [AXI 05], Appendix A2.4), the presence of viscous tangential stresses is very necessary to produce, or to destroy, vorticity. As a corollary, inviscid fluid dynamics deals with potential flows solely. Potential flows are conveniently described in terms of the velocity-potential Φ such that: V = grad Φ [1.51]

To conclude this subsection, it is worthwhile to emphasize the following important features of the Navier-Stokes equations: 1. Nonlinearity arises in the mass flux term of the mass equation due to fluid compressibility. It also arises in the momentum equation as a consequence of the momentum flux tensor. 2. The highest differential order occurs in the viscous operator. Consequently, when viscosity is disregarded, the differential order of the momentum equation is lowered and so is the number of boundary conditions to be fulfilled. 3. The nonlinearity of P , combined with the high differential order of the viscous operator, results in an extraordinary complexity of the solutions of the Navier-Stokes equations. Concretely, if the magnitude of the fluid velocity field is small enough, the solution is a regular (laminar) flow, which can be analysed as a deterministic process. But as soon as velocity becomes sufficiently large, the well behaved laminar regime becomes unstable and the flow becomes extraordinarily complex, being marked by the presence of disordered fluctuations, practically on any space and time scales. The phenomenon is called turbulence. Turbulent flow regimes are usually described by having recourse to the theoretical tools of random processes analysis. This aspect of the problem will be tackled in Volume 4, in relation to the random vibrations excited by turbulent flows. As shown in the next section, it is a straightforward task to linearize the Navier-Stokes equations and the fluid boundary conditions about a quiescent state. In contrast, it is a much more arduous task to linearize the momentum equation about a steady flow, since it is necessary to distinguish between the small fluctuations induced by the vibration and those induced by the turbulence. Furthermore, the latter are not necessarily small with respect to the permanent quantities. An additional difficulty lies in the fact that the fluctuating and permanent components of the flow are coupled together by P . Due to such a coupling, the very nature of the fluid-structure interaction problem is deeply modified in comparison with the quiescent case. In particular, the coupling between the permanent and the fluctuating flow velocity fields can induce a net input of mechanical energy from the permanent flow towards the vibrating structure. As a result, the vibration level increases rapidly up to quite unacceptable levels. The

26


phenomenon of flow induced instability, often termed fluid-elastic instability, will be studied in Volume 4. 4. As in the case of structural modelling, complex fluid problems can often be modelled using 1D or 2D formulations through suitable simplifying assumptions, as will be shown in the following chapters of this book. 1.3. Linear approximation of the fluid equations In most applications described in the present book, fluid oscillations about a stagnant state are of small magnitude, so it is assumed that the equations can be linearized about that state of static equilibrium. Of course, this assumption greatly simplifies the problem of solving the fluid equations. It can be safely adopted so far as viscous effects are neglected. Some care is however needed to model viscous dissipation, especially in the case of bluff bodies oscillating in an external fluid, because boundary layer separation can occur even at rather small oscillating flow velocities. 1.3.1

Linearized fluid equations about a quiescent state

1.3.1.1 Linear Navier-Stokes equations In order to analyse the small oscillations of a fluid about a state at rest, it is convenient to separate first the static and the fluctuating components of the field variables as follows: P( r ; t ) = P0 ( r ) + p( r ; t ) ρ ( r ; t ) = ρ0 ( r ) + ρ ( r ; t ) [1.52] V ( r ; t ) = V0 ( r ) + X f ( r , t ) = X f ( r , t ) e e e f ( ) ( r ; t ) = f 0( ) ( r ) + f ( ) ( r ; t ) Once again the subscript ( 0 ) is used to specify that we refer to the value in still fluid of the associated quantity. Since in linear theory the mean values of the fluctuating quantities are always zero, the quantities determined in still fluid may also be understood as mean values. As the thermodynamic changes are still discarded here, temperature is assumed to remain constant. The static pressure P0 is governed by the equilibrium equation: ( e ) gradP0 = f 0 [1.53] The fluctuating density ρ and pressure p are supposed to be small with respect to the mean values ρ0 and P0 . They are related to each other by the linear elasticity law [1.32]. Thus the small oscillations of the fluid are found to be governed by the following set of linear equations:


27

p = ρ c02

e ∂ ρ ∂ m( ) + ρ 0 div X f = ∂t ∂t ⎛ ⎞ ρ 0 X f + grad p − μ0 ⎜ ΔX f + 13 grad divX f ⎟ = f ( e ) ( r ; t ) ⎝ ⎠

(

[1.54]

)

1.3.1.2 The linear Euler equations To discuss further the basic properties of the system [1.54], it is convenient to start by making the following simplifying assumptions: 1. Here, the interest is focussed on the response properties of the fluid, that is on the left hand-side of the dynamic equations. As a consequence the external source terms arising in the mass equation [1.14] and the momentum equation [1.16] are assumed to be zero. Such source terms will be discussed in Chapters 4 and 5 in relation with the description of the forced sound waves. 2. As demonstrated later, discarding fluid viscosity is very convenient from a mathematical viewpoint and sufficient for explaining most of the physical aspects of the fluid oscillations about a quiescent state, except of course viscous dissipation. Consequently, the fluid will hereafter be assumed to be inviscid, at least up to Chapter 7 where viscous dissipation will be considered. 3. In reality, ρ0 , c0 can vary with r if the fluid is not homogenous. However, in this book presentation is essentially restricted to the case of homogeneous fluids, with a very few exceptions which will mentioned explicitly. In accordance with the first two assumptions made just above, the system [1.54] simplifies into the linear Euler equations, written here as: p = ρ c02

∂X f ∂ρ + ρ 0 div = 0 ⇔ ρ + ρ 0 div X f = 0 ∂t ∂t 2 ∂ Xf + grad p = 0 ρ0 ∂t 2

[1.55]

1.3.1.3 The sound wave equation in terms of a single field Starting from the Euler equations [1.55], the mass equation can be used together with the elastic law in order to eliminate either the pressure or the displacement field. This results in a sound wave equation of the second differential order, which

28


is expressed in terms of a single field variable. Let us start by expressing it in terms of X f . At first, ρ is eliminated between the mass equation, integrated once with respect to time, and the law of elasticity. Then, substituting the last expression for the pressure into the Euler momentum equation, the following wave equation is obtained: ρ 0 X f − grad ⎡⎣ ρ 0 c02 div X f ⎤⎦ = 0 [1.56] It is noticed that up to here the assumption of an homogeneous fluid is not needed to obtain a simple and compact wave equation, which of course is identical to that of the dilatational waves in an elastic solid (cf. [AXI 05], Chapter 1). However, to deal with fluid-structure coupled problems it is found more convenient to use a formulation in terms of pressure than in terms of displacement. This can be achieved by eliminating first ρ from the mass equation using the elastic law. Then, one differentiates the mass equation with respect to time and takes the divergence of the momentum equation. Provided the fluid is homogeneous, elimination of the term ρ div X leads to the simple and compact wave equation: 0

f

Δp −

1 p=0 c02

[1.57]

NOTE – As presented in many textbooks devoted to acoustics, the sound wave equation can also be expressed in terms of velocity-potential leading, in homogeneous fluid, to the same form as [1.57]. The same is true, if a displacement instead of a velocity potential is used.

1.3.2

Linearized boundary conditions

1.3.2.1 Fluid-structure coupling term at a wetted wall On a wetted wall (W ) the interface condition differs depending on whether the fluid is modelled as viscous or not. According to the viscous model, the fluid must adhere to the wall; thus the appropriate condition is: X f − Xs = 0 ⇔ X f − X s = 0 ⇔ X f − X s =0 [1.58] (W )

(W )

(W )


29

Figure 1.13. Fluid structure interface

According to the non viscous model, the fluid can slide freely along the wall, but it is also assumed to keep contact with it, thus the condition is: X f − X s .n = 0 ⇔ X f − X s .n = 0 ⇔ X f − X s .n =0 [1.59]

(

)

(W )

(

)

(W )

(

)

(W )

where n is the unit normal vector to (W ) oriented conventionally from the

structure towards the fluid, as indicated in Figure 1.13. Again, according to the hypothesis of small motions, (W ) refers to the static configuration. By using the momentum equation [1.55], the condition [1.59] can be expressed in terms of the normal pressure gradient at the wall instead of the fluid normal acceleration: ∂p X f − X s .n = 0 ⇔ grad p.n = = − ρ 0 X s .n [1.60] (W ) ∂n (W ) (W ) (W )

(

)

As a particular case, at a fixed wall the gradient of the pressure is found to vanish in the normal direction, which is equivalent to saying that the mass flux through the fixed wall vanishes. 1.3.2.2 Free surface of a liquid in a gravity field As already mentioned in section 1.1, gravity and surface tension add some potential energy to the free surface of a liquid. In this subsection we consider solely the effect of gravity. The appropriate linearized boundary condition can be established using a few distinct approaches. The most intuitive and less formal manner is to consider a vertical displacement Z ( x, y , H ; t ) of the liquid as reckoned from the reference level z = H, and to calculate the change of static pressure at that level, see Figure 1.14. At static equilibrium, configuration of the free surface is denoted (Σ 0 ) and the pressure field is readily found to be:

30


P0 (z) = Pa + ρ 0 g(H − z)

[1.61]

Pa denotes the gas pressure.

Figure 1.14. Free surface oscillating in a permanent gravity field

Provided the volume of the gas is practically infinite, a small displacement of the liquid does not change Pa . Thus, if Z is positive the perturbed pressure is given by the static equilibrium condition: P ( x, y , z; t ) = Pa + ρ 0 g(H + Z ( x, y , H ; t ) − z)

[1.62]

The fluctuating pressure at (Σ 0 ) is thus found to be: p ( x, y , H ; t ) = P ( x, y, H ; t ) − P0 (H) = ρ 0 gZ ( x, y, H ; t )

[1.63]

In agreement with the concept of mechanical impedance already introduced in the context of structural mechanics, the condition [1.63] can be interpreted as an elastic impedance, since it relates a stress component to a displacement component through a law of proportionality. On the other hand, the condition [1.63] can also be derived based on a variational calculus. The gravity potential of the fluid in the deformed configuration of the free surface is written as: ⌠ ⎮

Ep = ⎮

⎮ ⎮ ⌡( Σ 0 )

⌠

dxdy ⎮⎮

H +Z

⌡0

ρ0 gz dz =

ρ0 g ⌠⎮ ( H 2 + +2ZH + Z 2 ) dxdy 2 ⎮⎮⌡( Σ 0 )

[1.64]

This result can be further simplified by omitting the constant and the linear terms. The first simplification occurs because a potential can always be defined save on an additive time function or constant. Thus the constant term within the brackets can be removed. The second simplification occurs as a mere consequence of the fluid incompressibility which prevents any change of fluid volume. Thus the linear term


31

within the brackets vanishes. Therefore, the potential energy related to the motion of the free-level is finally written as: Ep =

1⌠ ⎮

2⎮ ⌡( Σ 0 )

ρ 0 gZ 2 ( x, y , H ; t )dxdy

[1.65]

The virtual work of pressure on ( Σ 0 ) is: ⌠

δWp = ⎮⎮

( )

⌡ Σ0

pδ Z ( x, y, H ; t )dxdy

[1.66]

The Lagrangian of the superficial fluid is thus written as: L=

⌠ ⎮ ⎮ ⎮ ⎮ ⎮ ⌡ Σ0

( )

⎛ 1 ⎞ 2 ⎜ − ρ0 gZ +Wp ⎟ dxdy ⎝ 2 ⎠H

[1.67]

It is worth to emphasize that the energies entering into the Lagrangian are calculated based on the non deformed configuration (Σ 0 ) and, even more important, it is tacitly assumed that their analytical form is independent of the sign of Z, which is a requirement inherent in any stationary principle as seen in [AXI 04], Chapter 3. Applying the principle of least action, equivalent here to the principle of stationary potential energy, with the aid of [1.66], the following condition is readily obtained: ⌠

δ ⎡⎣ L ⎤⎦ = − ⎮⎮ ⎮

( )

⌡ Σ0

( ρ gZ − p ) H δ Zdxdy = 0 0

∀δ Z admissible

which is equivalent to the condition [1.63]. It can be expressed in terms of pressure solely by using the vertical component of the momentum equation for an inviscid fluid, which gives: ρ0 Z f +

∂p ∂z

=0

[1.68]

H

Differentiating twice the relation [1.63] with respect to time, we arrive at the free surface condition written in terms of pressure: p ( x, y , H ; t ) + g

∂p ∂z

=0

[1.69]

H

Validity of the above approaches to derive the boundary condition [1.63], or [1.69], can be questioned because they are supposed to hold whatever the sign of Z may be. However, it can be rightly objected that if Z is negative, the actual pressure at H becomes equal to the gas pressure Pa and not to the value prescribed by [1.63],

32


nor to [1.69]. It is thus of interest to check that [1.63] and [1.69] stand for the linearized approximation of the actual boundary condition which is nonlinear in nature. On the actual configuration ( Σ L ) of the free surface, defined as the still unknown function z = Z 0 ( x, y; t ) , the fluid particles must comply with two distinct conditions. The first is a kinematical constraint which serves to characterize those fluid particles which are located on ( Σ L ) . Denoting Z p ( x, y, z; t ) the vertical position of the particles in Eulerian coordinates, those which are on ( Σ L ) are such that: Z p ( x , y , z; t ) − Z 0 ( x , y ; t ) ≡ 0

[1.70]

It follows that the vertical velocity of the superficial particles must be equal to the time derivative of the free surface z = Z 0 ( x, y; t ) . Whence the kinematical nonlinear relation: DZp Dt

=

dZ + V .grad Z p = 0 = Z 0 ∂t dt

∂Z p

[1.71]

where Z p is the vertical component of the velocity of a particle at the free surface and V is the Eulerian velocity field. The linear version of [1.71] simply states that the vertical component of the fluid velocity, (Eulerian or Lagrangian, indifferently), are also equal to the time derivative of the free surface. The second condition to be fulfilled is that of dynamical equilibrium. As a potential flow model is adopted, by using [1.43] and [1.49] the appropriate momentum equation is readily shown to be: ∂Z ∂Φ Z 0 = p = ∂t ∂z

[1.72]

Φ is the velocity potential defined by relation [1.51].

The second condition to be fulfilled is that of dynamical equilibrium of the superficial fluid particles. Since a potential flow model is adopted, by using [1.43] and [1.49], the appropriate momentum equation is found to be: ⎛ ∂ V 1 2 ⎞ ρ0 ⎜ + gradV ⎟ + gradP + ρ 0 g = 0 [1.73] ⎝ ∂t 2 ⎠ Notice that the permanent body force due to gravity must be accounted for in [1.73] because P stands for the total pressure field, including both the static and fluctuating components (see [1.52]). The vertical component of [1.73] reads as: ρ0

ρ ∂ Vz ∂ ⎛ ⎞ + ⎜ P + 0 (Vx2 + V y2 + Vz2 ) + ρ 0 gz ⎟ = 0 ∂ t ∂z ⎝ 2 ⎠

[1.74]


33

With the aid of Φ , equation [1.74] is further transformed into: ⎞ ∂ ⎛ ∂Φ ρ + P + 0 (Vx2 + V y2 + Vz2 ) + ρ 0 gz ⎟ = 0 ⎜ ρ0 ∂t 2 ∂z ⎝ ⎠

[1.75]

Integration of [1.75] is immediate, producing the unsteady Bernoulli equation: ρ0

∂Φ 1 + P + ρ 0 (Vx2 + V y2 + Vz2 ) + ρ 0 gz = C ( t ) ∂t 2

(

)

[1.76]

Furthermore, the time function C ( t ) can be set to zero without loss of generality since Φ is defined save on an arbitrary time function. At the free surface, z is equal to Z 0 ( x, y; t ) and P is equal to the atmospheric pressure, assumed to be constant. Thus, it turns out that the dynamical condition can be written as: ⎡ ∂ Φ 1 ⎛ ⎛ ∂ Φ ⎞2 ⎛ ∂ Φ ⎞2 ⎛ ∂ Φ ⎞2 ⎞ P ⎤ a ⎢ ⎥ ⎟ + ⎜⎜ + + + + =0 gz ⎟ ⎜ ⎟ ⎜ ⎟ ⎢⎣ ∂ t 2 ⎜⎝ ⎝ ∂ x ⎠ ⎝ ∂ y ⎠ ⎝ ∂ z ⎠ ⎟⎠ ρ 0 ⎥⎦ z = Z0

[1.77]

Linearizing [1.77] yields: ⎡ ∂ Φ Pa ⎤ =0 ⎢ ∂ t + ρ + gz ⎥ 0 ⎣ ⎦ z = Z0

[1.78]

Then, the condition [1.78] is derived with respect to t to eliminate Pa and Z 0 using the kinematical condition [1.72]. The result is written as: ∂ 2Φ ∂Φ +g 2 ∂t ∂z

=0

[1.79]

z=H

Conditions [1.79] and [1.69] are of the same type. Moreover, the vertical momentum equation can be used to shift from one formulation to the other: ρ0

∂ 2Z ∂ p ∂ 2Φ ∂ p ∂Φ + = 0 ⇔ ρ + = 0 ⇔ ρ0 + p=0 0 ∂t 2 ∂z ∂t∂z ∂z ∂t

[1.80]

where p is the fluctuating component of pressure in agreement with [1.52]. Here also when integrating the momentum equation with respect to z, the constant can be set to zero without loss of generality. From the relations [1.80], it follows immediately that: ∂ 3Φ 1 ∂2p = − ∂t 3 ρ 0 ∂t 2

;

∂ 2Φ 1 ∂p =− ∂t∂z ρ 0 ∂z

[1.81]

Deriving [1.79] with respect to time and using [1.81], condition [1.69] is recovered as desired, which validates the consistency with the linearization procedure.

34


1.3.2.3 Surface tension at the interface between two fluids As in the case of solids, cohesion of liquids results from attractive intermolecular forces. The tendency of such forces is to enclose a finite part of liquid by a surface, which acts as an elastic membrane to maintain a configuration minimizing the potential energy. The phenomenon is the cause, among others, of the formation of liquid droplets and gas bubbles. Furthermore, the molecules of a liquid can also interact with those of another liquid, in such a manner that the cohesive forces can be very sensitive to the chemical nature of the fluids in contact and even to a small change in the chemical composition of one of them, as it can be verified by adding a small amount of soap, or any other surfactant substance to water, which lowers surface tension by a large amount.

Figure 1.15. Experimental set-up for demonstrating surface tension

Surface tension at the interface between two non miscible fluids can be easily brought in evidence by performing the elementary experiment shown in Figure 1.15. The device is a U-shaped metallic wire and a thread of cotton fixed to the ends of the U branches. The length of the thread is larger than the width of the U in such a way that it hangs under the effect of its own weight, as shown in the left-hand side picture. Then, a thin liquid film is formed by immersing the frame into a liquid, typically a soap solution, and removing it. In passing, the reason for adding a surfactant to water is that surface tension of plain water is too strong for liquid film or bubbles to last for any length of time. As observed in the right-hand side picture, the thread is tensioned, materializing in a plane curve of negative curvature. Careful observation shows that it is shaped as a circular arc. From such an experimental fact, it can be deduced that the tension forces per unit length are uniform and radial and that the liquid film area is minimized, in agreement with the solution of a famous isoperimetric problem, broadly known as the Dido Problem. According to the legend, Queen Dido of Carthage landed as a refugee from the Phoenician city of Tyria on the African north coast, at the spot which will be the seat of the future cities of Carthage and then Tunis. She asked the local king named Jarbas to allow her to get settled there and to found a city with her companions. Jarbas was certainly


35

not devoid of any sense of hospitality and humour, but his generosity was rather minimal. So he offered Dido a piece of the land which could be enclosed by the hide of a bull. Minimal generosity calls for maximal trick in return, as Dido showed brilliantly. Indeed, history tells that she cut the bull skin in very thin thongs to form a semi-circle using the African coast as a supplementary boundary, solving thus the optimisation problem raised by the greediness of Jarbas and marking the birth of Carthage, circa 850 B.C.

Figure 1.16. Idealized geometry of the system

The Dido problem is classically used to illustrate the efficiency of the Lagrange multiplier technique to solve optimisation problems, see [AXI 04], Chapter 4. As far as the present problem is concerned, we need to determine the maximum area under the curve z(x) materialized by the thread, in such a way that the area of the film is minimized, x varying from zero up to / 2 due to the symmetry of the U frame, see Figure 1.16. The constraint condition is: /2

⌠ ⎮ ⎮ ⌡o

1 + z ′2 dx =

L 2

[1.82]

L/2 is half length of the thread and / 2 is half the width of the U frame. The constrained Lagrangian is written as: /2

L=

⌠ ⎮ ⎮ ⎮ ⎮ ⌡o

⎛ ⎛L 2 ⎞⎞ ⎜ z ( x ) − Λ ⎜ − 1 + z ′ ⎟ ⎟ dx ⎝ ⎠⎠ ⎝

[1.83]

The Euler-Lagrange equation follows as: d ⎛ dL ⎞ dL + Λ z ′′ = −1 = 0 ⎜ ⎟− dx ⎝ dz ' ⎠ dz (1 + z ′2 )3/ 2

[1.84]

36


In the coefficient of the Lagrange multiplier we recognize the curvature of the line z(x). Thus it is readily found that the solution of [1.84] is necessarily a circular arc of radius R and Λ = R. To determine the value of R which satisfies the constraint condition about the length of the arc, the circle equation is written as: x 2 + ( z − zc ) = R 2 2

[1.85]

where the centre of the circle is at x = 0 and z = zc . With the aid of [1.85], it is straightforward to derive the following mathematical relations: zc = ± R 2 − ( / 2 ) ; z ′2 = 2

x2 R − x2 2

[1.86]

Using [1.86], the constraint condition [1.82] is transformed into the definite integral: / 2R

⌠ ⎮ ⎮ ⎮ ⎮ ⌡o

du 1− u

2

=

L 2R

[1.87]

Whence the equation which gives implicitly the radius of the circle: ⎛ ⎞ L arcsin ⎜ ⎟= ⎝ 2R ⎠ 2R

[1.88]

It may be noticed that two arc lengths lead to the same value of R, namely the solution L of the equation [1.87] and the supplementary arc L′ = 2π R − L . By varying the geometry of the frame, the present experiment is suitable to prove that the surface tension T is proportional to L and independent of the film area. Furthermore, it could also be shown that it is also independent of the thickness of the film, as indicated by the observation that the radius of a soap bubble remains constant up to the time the bubble pops due to the evaporation of the envelope. Incidentally, a good indicator of evaporation is the change in the interference colours as the film is thinning down. Hence it is suitable to define the surface tension per unit length and for a single interface as: σf =

T 2L

[1.89]

where σ f is termed the capillary force per unit length. The value of it depends on the chemical nature of the fluids in contact and is sensitive to temperature. For instance, at 20°C, σ f = 0.0726 N/m in the case of a water/air interface, and 0.465 N/m, in the case of a mercury/air interface. Temperature dependence of capillary force of plain water is illustrated in Appendix A2, Table A2.4.


37

As a preliminary to establish the capillary condition at a slightly deformed interface between two fluids, the material law [1.89] is applied first to the static equilibrium of a soap bubble. We consider an infinitesimal element of the curved liquid film, described by using curvilinear orthogonal coordinates α , β , see Figure 1.17.

Figure 1.17. Static equilibrium of a curved liquid film

The area of the element is dS = Rα Rβ dαdβ where Rα , Rβ denote the main curvature radii of the surface. Tα = 2σ f Rα dα and Tβ = 2σ f Rβ d β are the capillary forces exerted at the boundaries of the element, in the tangential directions. The equilibrium in the normal direction implies necessarily a pressure discontinuity δP = P2 − P1 across the film to satisfy the normal force balance: δ PdS = 2σ f ( Rα + Rβ ) dα d β

Whence the relation, broadly known as the Laplace capillary law: ⎛ 1 1 ⎞ δ P = 2σ f ⎜ + ⎟⎟ ⎜R ⎝ α Rβ ⎠

[1.90]

A soap bubble floating in air is shaped as a sphere, which provides the smallest possible area for a given volume and when the internal gas of two soap bubbles are put in communication, it is observed that the smallest bubble empties itself in the biggest, because in agreement with the Laplace law, pressure should be higher in the smaller bubble. For a liquid drop, or a gas bubble immersed in a liquid, the factor 2 σ f is to be replaced by σ f since there is only one interface. In problems involving also the gravity field, it is found useful to define the capillary length α f as: αf =

σf ρf g

[1.91]

38


Generally, α f is of the order of a few millimetres and even less. For instance in a water/atmospheric air interface at normal conditions α f is about 2.6 mm. As a classical application, a liquid is sucked into a capillary tube of radius r0 (typically less than one millimetre) up to a height given by the equilibrium equation: h ρ f gπ r02 = 2π r0σ f cos θ ⇒ h =

2α 2f cos θ r0

[1.92]

where θ is the contact angle of the meniscus with the wall, see Figure 1.18. Flatness of the free surface of the liquid in the tank, except in the immediate vicinity of the walls, is a mere consequence of the fact that the scale length is much larger than r0 .

Figure 1.18. Water column in a capillary tube

The Laplace law is easily particularized to the case of a slightly deformed interface between two fluids, as sketched in Figure 1.19 in the case of a wavy free surface with characteristic wavelength λ. Indeed, it suffices to substitute into [1.90] the linear curvatures of the deformed interface Z(x,y,H). Therefore, using Cartesian coordinates, the linear capillary condition is written as: ⎛ ∂ 2Z ∂ 2Z ⎞ p(x, y,H) = −σ f ⎜ + = −σ f ΔZ ( Σ ) ⎟ 2 0 ∂ y 2 ⎠ (Σ ) ⎝∂ x 0

[1.93]

Z ( x, y , z; t ) denotes the vertical displacement of the liquid and the Laplacian is

restricted to the coordinates of the non deformed interface ( Σ 0 ) assumed here to be a horizontal plane due to gravity. The density per unit area of capillary potential is: 1 ec = σ f 2

⎛ ⎛ ∂ Z ⎞2 ⎛ ∂ Z ⎞2 ⎞ ⎜⎜ ⎟ + ⎜ ⎝ ∂ x ⎟⎠ ⎜⎝ ∂ y ⎟⎠ ⎟ ⎝ ⎠ (Σ0 )

[1.94]


39

It may be noted that the potential density [1.94] is similar to that of a stretched membrane in transverse displacement, provided σ f is replaced by the in-plane stretching force exerted at the contour of the membrane and that shear components are discarded. Gravity also being taken into account, the linear condition to be fulfilled on a free surface separating a liquid and a practically infinite volume of gas (free atmosphere for instance) is obtained by adding the effects of gravity and surface tension: ⎛ ∂ 2Z ∂ 2Z ⎞ p(x, y,z) + σ f ⎜ 2 + − ρ 0 gZ 2 ⎟ ⎝∂ x ∂ y ⎠

=0

[1.95]

H

Figure 1.19. Wavy free surface

1.3.3

Physical quantities and oscillations of the fluid

From what precedes, it results that the quiescent state used as a reference to describe the small fluid oscillations is entirely characterized by five distinct physical quantities, controlling five distinct coupling mechanisms between the vibration of the fluid and that of the solid. It is of interest to recapitulate them and to define a few dimensionless parameters which allow one to determine their relative importance, based on the equilibrium equations derived in the last section. In reality, the fluid motion induced by the vibrating structure results, of course, from all the coupling mechanisms operating together. So it is appropriate, for mathematical convenience at least, to identify carefully those which can be considered as negligible in relation to the specificities of the problem to be modelled. In Appendix A2, the reader will find an illustrative data set of the mechanical properties of several common materials. 1.3.3.1 Mean value of fluid density The mean value of fluid density governs the inertial effect exerted by a certain mass of fluid set into motion by the vibrating solid. It may be accounted for by adding the kinetic energy of the fluid to that of the solid, or equivalently by mass

40


coefficients added to the structure. The relative importance of fluid inertia in comparison with the other fluid effects can be suitably characterized by performing a dimensional analysis of the fluid equations and by comparing the relative magnitude of the terms related to the five quantities mentioned just above. The pressure field which gives rise to fluid inertia is governed by the equations [1.57] in which compressibility is discarded and by the wall condition [1.60], leading to the system: Δp = 0 grad p.n

(W )

= − ρ 0 X s .n

[1.96] (W )

Let us designate by Lx , Ly , Lz the characteristic length scales of the pressure field in the Ox, Oy and Oz directions, respectively. According to the first equation [1.96], the relative magnitudes of the pressure gradients in the Ox, Oy and Oz directions are given by: 1 ⎛ ∂p ⎞ 1 ⎛ ∂p ⎞ 1 ⎛ ∂p ⎞ ≈ ⎜ ⎟⇒ ⎜ ⎟≈ Lx ⎝ ∂x ⎠ Ly ⎜⎝ ∂y ⎟⎠ Lz ⎝ ∂z ⎠ ∂p ⎛ ∂p ⎞ Ly ∂p ⎛ ∂p ⎞ Lz ≈⎜ ⎟ ≈⎜ ⎟ ; ∂y ⎝ ∂x ⎠ Lx ∂z ⎝ ∂x ⎠ Lx

[1.97]

The symbol ‘ ≈ ’ means that the related quantities are of the same order of magnitude. The pressure gradients can be related to the vibration of the solid by using equation [1.60]. Assuming that the wetted wall (W ) vibrates at the characteristic pulsation ω with a displacement amplitude X s in the Ox direction perpendicular to (W ) , it follows that: ∂p ≈ ω 2 ρ0 X s ∂x

;

L ∂p ≈ ω 2 ρ0 X s y ∂y Lx

;

∂p L ≈ ω 2 ρ0 X s z ∂z Lx

[1.98]

Using the equation [1.15] and the momentum equation [1.55] for an incompressible fluid, it is easily shown that: X f ≈ Xs

; Yf ≈ X s

Ly Lx

; Zf ≈

Lz Lx

[1.99]

where X f , Y f , Z f are the amplitude of the fluid oscillations in the Ox, Oy and Oz directions, respectively. Hereafter the scale length ratios appearing in [1.98] and [1.99] are referred to as confinement ratios of the fluid. As a gross order of magnitude, the kinetic energy of the oscillating fluid may be written as:

Introduction to fluid-structure coupling t2 2 ⎛ ⎛ L ⎞2 ⎛ L ⎞2 ⎞ 1 ⌠⎮ 1 ρ0 ⎮ X f .X f dV ≈ ρ 0Vf X s ⎜ 1 + ⎜ y ⎟ + ⎜ z ⎟ ⎟ ⎜ ⎝ Lx ⎠ ⎝ Lx ⎠ ⎟ 2 ⎮⌡(Vf ) 2 ⎝ ⎠

41

[1.100]

The result [1.100] points out that besides to be proportional to the mean fluid density, the inertia effect is also proportional to the squared confinement ratios. As further discussed in Chapter 2, it turns out that due to the assumption of fluid incompressibility, the values of Lx , Ly , Lz are governed by the geometry and the boundary conditions of the fluid volume (Vf ) . In particular, it will be shown that inertia effects can be very large if the fluid is highly confined by solid walls. 1.3.3.2 Gravity field Discarding for a while the surface tension effect, the gravity field specifies the vertical direction Oz and governs the shape of the free surface of a liquid at the static state of equilibrium, which is horizontal. Furthermore, as shown in subsection 1.3.2.2, it governs also the fluctuating pressure related to the small vertical oscillations of the free surface. The form [1.69] is found convenient to assess the relative importance of gravity to inertia in a vibratory problem. Dimensional analysis of [1.69] leads to define the so-called oscillatory Froude number as: F =

ω 2 Lz g

[1.101]

Gravity is significant with respect to inertia in the range F ≤ 1 . At the opposite, in the range F >> 1 , it can be discarded and a node of pressure can be assumed at the free surface: p ( x, y , H ; t ) = 0

[1.102]

In the earth’s gravity field, assuming Lz 1 m , coupling becomes negligible at frequencies larger than about 1 Hertz. 1.3.3.3 Surface tension As shown in subsection 1.3.2.3, surface tension leads also to an elastic impedance term at a free surface, or at the interface between two distinct immiscible fluids. The relative importance of surface tension to gravity can be readily inferred from the condition [1.95]. The order of magnitude of the small curvature of the wavy surface is: ∂ 2Z ∂ 2Z Z ≈ ≈ ∂ x 2 ∂ y 2 L2

[1.103]

42


where it is assumed that Lx ≈ Ly ≈ L for the sake of simplicity. Substituting [1.103] into [1.95], we are led to the conclusion that surface tension coupling is negligible in comparison with gravity if: αf L

>1 surface tension is negligible with respect to inertia. 1.3.3.4 Fluid elasticity Due to compressibility, the fluid behaves as an elastic continuous medium. Small elastic oscillations of a homogeneous fluid were found to be governed by the conservative linear wave equation which can be expressed either in terms of fluid displacement or in terms of pressure. Dimensional analysis of either equation [1.56], or [1.57], leads to the definition of the oscillatory Mach number: Λa =

ωL L = = ka L λa c0

[1.107]

λa is the wavelength and ka = 2π / λa the wave number of the acoustical waves. In

agreement with relations [1.107] Λ a can also be interpreted as a reduced acoustical wave number. Hereafter, Λ a will also be referred to as a compressibility parameter to emphasize that fluid compressibility can be neglected with regard to inertia in the


43

domain Λa 1 , inertia forces are much larger than viscous forces and dissipation due to fluid viscosity is small. As a consequence, besides depending upon the kinematic viscosity ν 0 , viscous dissipation largely depends on the smallest length scale of the fluid volume. This is not surprising, as in laminar flows, viscous forces are proportional to the gradient of the flow velocity (cf. Newton’s law of friction [1.35]) and flow velocity is proportional to the confinement ratio, cf. relation [1.99].

Chapter 2

Inertial coupling

The effect of the presence of a fluid on the natural frequencies and mode shapes of relatively flexible and not too compact structures interacting with a dense fluid like water can be significant, and even of paramount importance. In such cases, the main physical mechanism to be studied is inertial coupling, which is the subject of the present chapter. As will be seen, in linear systems fluid inertia can be modelled as an added mass matrix operating on the degrees of freedom of the structure, which means that no additional degree of freedom related to the fluid is required. Furthermore, the coefficients of the added mass matrix reflect important physical features of inertial coupling, which is conservative in nature, highly directional and sensitive to boundary conditions. Such features can be made apparent by solving a few analytical problems, either of discrete nature or in terms of continuous formulations. This allows one to understand, in particular, how the vicinity of a free surface or, at the opposite extreme, of a fixed wall, near a vibrating structure can greatly modify the inertial effects and why the mode shapes of a structure can be affected by fluid inertia. Also, conditions are established under which lower dimension (1D and 2D) fluid models may be sometimes devised as simplified alternatives to the original 3D fluid field. The information gained by the analysis of such pedagogical and somewhat academic examples is very useful as a guide line at least, to study real systems of practical interest. Actually, it is worth emphasizing that availability of analytical or semi-analytical solutions rapidly decreases as the complexity of the fluid-structure coupled system increases. Furthermore, numerical studies of fluid-structure interaction problems using either finite element (FEM) or boundary element methods (BEM) are significantly more delicate and heavier than those of structures in vacuum. The presentation of the basic principles of FEM is postponed to Chapter 6, that of BEM is however beyond the purview of this book.

46


2.1. Introduction In order to study the vibrations of a structure coupled to a dense fluid, it is appropriate to concentrate first on the inertial effects by disregarding the other coupling mechanisms. Therefore, the quantities 1/ c f , σ f ,ν f and g, introduced in Chapter 1, are first set to zero. The mechanisms thus neglected will be studied individually later in the present book, to show how they interact with fluid inertia and the vibrations of the structure.

Figure 2.1. Fluid-structure coupled system

Thus, in the present chapter the fluid is entirely characterized by its density which will be hereafter denoted ρ f while with that of the solid will be denoted ρ s . Starting from equations [1.6] and [1.96], the fluid-structure coupled system is written as: M s ⎡⎢ X s ⎤⎥ + K s ⎡⎣ X s ⎤⎦ = − pnδ ( r − r0 ) ⎣ ⎦ [2.1] Δp = − ρ f X s .nδ ( r − r0 ) where δ denotes the Dirac distribution used to concentrate the fluid-structure coupling terms at the fluid-structure interface, that is at current position r0 on the vibrating wetted wall (W ) , see Figure 2.1. The right-hand side of the first

equation [2.1] means that the structure is loaded by the fluctuating pressure exerted on the wall and the right-hand side of the second equation means that the fluid is loaded by the motion of the wetted wall. As these terms are related to the direction

Inertial coupling

47

normal to the wall exclusively, the coupling is highly directional. In particular, it vanishes for any tangential vibration. The equations [2.1] must be complemented by a suitable set of boundary conditions to describe the supports of the structure and the fluid boundaries other than vibrating walls. Concerning the fluid, at a fixed wall (W0 ) the boundary condition is: X f .n

(W0 )

= 0 ⇔ grad p.n

(W0 )

∂p = ∂n

(W0 )

=0

[2.2]

In the case of a liquid separated from a large volume of gas by a free surface ( Σ 0 ) , the following condition holds: p (Σ ) = 0 0

[2.3]

Inspecting equations [2.1], it is noticed that, because of linearity, the pressure field is necessarily proportional to the normal acceleration of the wall; therefore the structure is loaded by a force which is also proportional to the normal acceleration of the wall. As already outlined in Chapter 1, two consequences can be immediately derived. First, the force exerted by the fluid is inertial in nature, then pressure is an auxiliary variable which can be eliminated. In other words, the incompressible fluid does not bring any additional degree of freedom (in short, DOF) to the system. As a third consequence, the coupled system [2.1] can be solved as a modal problem by considering harmonic vibrations, governed by the following equations, where (C.B.C) stands for the conservative boundary conditions verified by the structure and the fluid: K s ⎡⎣ X s ⎤⎦ − ω 2 M s ⎡⎣ X s ⎤⎦ + pnδ ( r − r0 ) = 0 Δp − ω 2 ρ f X s .nδ ( r − r0 ) = 0 [2.4] +(C.B.C)

Since both structure and fluid are modelled here as conservative and stable dynamical systems, it can be anticipated that the natural modes of [2.4] have the same properties as those already identified in the case of solids, namely, the natural frequencies are positive, eventually null, the mode shapes are real and constitute an orthogonal vector basis to expand the solution of any forced problem as a modal series. Depending on the particularities of the problem, various mathematical techniques can be used to solve it. The object of this chapter is to solve a few generic examples by using the analytical methods already described in [AXI 04, 05], and to describe the salient physical features of the inertial coupling in relation to the geometry and the boundary conditions of the fluid. Section 2.2 deals with the simplest case, where the displacement field of the structure and that of the fluid can be easily anticipated. Such problems are discrete in nature as they can be formulated directly as a set of algebraic and linear equations, avoiding thus the burden of

48


solving differential equations. In section 2.3, the problem is set in its general differential form and discretized by projecting equations [2.4] on the natural modes of vibration of the structure in vacuum. This technique provides the general result of theoretical interest that the fluid inertial coupling can be described as an added mass matrix which is self-adjoint and positive. As shown in a few illustrating examples, it can be used as an efficient tool for solving many problems of practical interest. Presentation of the finite element method is postponed to Chapter 6 where fluid compressibility and free surface effects are also taken into account. 2.2. Discrete systems 2.2.1

The fluid column model

The simplest conceivable fluid-structure systems comprise one or several springmass oscillators, connected to one or several columns of liquid contained in rigid tubes. They are suitable to highlight the physical aspects of the inertial coupling with minimum mathematical effort. Furthermore, they can be conveniently used to introduce the more sophisticated methods needed for dealing with continuous systems. The solid body of the oscillator is assumed to act as an water-tight piston in contact with the liquid. By definition, in a tube the volume offered to the fluid is characterized by one dimension L, termed the length, which is much larger than the two others, the so-called transverse dimensions, see Figure 2.2. The fluid column contained in a tube, or pipe, can be viewed as the fluid counterpart of a beam.

Figure 2.2. Geometry of a tube filled with a fluid

As indicated in Figure 2.2, the tube can be curved and the cross-sectional area offered to the fluid is not necessarily constant. The curvilinear abscissa along the tube is denoted s; by convention, it is taken as positive from (A) to (B). R ( s ) denotes the radius of the cross-section at s, supposed to be circular for convenience. The tube length L is assumed to be much larger than the tube radius:

Inertial coupling

L / R >> 1

49

[2.5]

As a structural element, a tube is modelled as a slender beam, see Chapter 6. However, here, it is assumed to be rigid and fixed. Concerning the fluid column, since the aspect, or slenderness, ratio [2.5] is large, the idea is to model it by adopting essentially the same simplifying assumptions as those used to model a structural element as a slender beam. Therefore, the 3D pressure and fluid velocity fields are approximated by 1D fields which vary along the longitudinal direction solely. To be more specific, the local fields are replaced by their mean values, as averaged over the cross-sections: p( s) =

1 ⌠⎮ p ( r ) dS S f ( s ) ⎮⌡( S )

X f ( s ) =

1 ⌠⎮ X f ( r ) dS S f ( s ) ⎮⎮⌡( S )

[2.6]

S f ( s ) = π R 2 ( s ) is the cross-sectional area of the fluid column at the curvilinear

abscissa s. Furthermore, it is found convenient to replace the mean flow velocity by the volume velocity q(s), defined as: q( s ) = S f ( s ) X f ( s ). [2.7]

is the unit vector normal to S f ( s ) , conventionally oriented from (A) to (B).

NOTE:

volume velocity versus mass flow

In principle, the mass flow q ( s ) = ρ f q ( s ) could be used as the kinematical variable, instead of the volume velocity q(s). Actually, so long as the mean value of the fluid density ρ f is the same everywhere within the pipe using q instead of q is slightly advantageous as it allows one to condense the mathematical formulas. However, the pertinence of choosing the volume velocity appears when dealing with heterogeneous fluids of distinct densities, as in the straight tube shown in Figure 2.3. At the interface between the two columns of fluid the following conditions of continuity must hold: p ( ) ( x0 ) = p − = p ( 1

2)

( x0 ) = p+

ρ 1 2 X (f ) = X (f ) ⇔ q− = q+ ⇔ q− = 1 q+ ρ2

[2.8]

50


q− = q+ p− = p+

ρ1

q−

q+

p−

p+

ρ2

Figure 2.3. Straight tube filled with two incompressible fluids of distinct densities

The pressure equation arises as a necessary condition for equilibrium of the mass-less interface. The kinematical equation is a direct consequence of fluid incompressibility. It may be noticed that the flow continuity is expressed in a more condensed manner by using the volume velocity than the mass flow. Finally, the same relations of continuity at the interface hold even if compressibility is accounted for. This is because there is no forces exerted on the interface, and no fluid lost or gained trough it. Such a result can also be justified by claiming that at the scale of the interface (infinitesimal length ε of tube extending from x0 − ε / 2 to x0 + ε / 2 ) compressibility of the fluid is negligible in any case. Indeed the oscillatory Mach number ωε / c0 , defined in Chapter 1, subsection 1.3.3.4, can be made as small as needed. As shown in Chapter 4, it turns out to be advantageous to use the volume instead of the mass flow also in acoustics. The left-hand side of the 3D linearized Euler equations for an incompressible fluid, deduced as a special case of equations [1.54], is integrated over S f ( s ) as follows: ⌠ ⎮ ⎮ ⎮ ⌡( S f ⌠ ⎮ ⎮ ⎮ ⌡( S f

∂q div X f ( r ) dS = ∂s )

)

(

∂q ∂p ρ f X f ( r ) + grad p . dS = ρ f + S f ( s) ∂t ∂s

)

[2.9]

where the momentum equation is projected on the longitudinal direction. Free harmonic oscillations of the fluid q ( s ) eiω t , p ( s ) eiω t are governed by the following one-dimensional equations:

Inertial coupling

dq =0 ds

51

[2.10]

dp iωρ f q + S f =0 ds

where the bar over the averaged pressure is dropped to alleviate the notation. Finally, q(s) is eliminated to produce the one-dimensional equation governing the fluctuating pressure: d ⎛ d p⎞ ⎜Sf ⎟=0 ds ⎝ ds ⎠

[2.11]

Hence, it is found that the same equations hold whether the tube is straight or curved. Either the volume or the mass flow is constant and, if S f is also constant, the pressure varies linearly along the tube. The one-dimensional approximation to deal with slender fluid columns shall be extended to the case of compressible fluids in Chapter 4 and its range of validity will be clarified in Chapter 5. 2.2.2

Single degree of freedom systems

2.2.2.1 Piston-fluid system: tube of uniform cross-section

Figure 2.4. Piston-fluid system 1 DOF

Figure 2.4 sketches the simplest conceivable fluid-structure system, which will be used through the present book as an archetype for introducing the various mechanisms of fluid-structure coupling. It comprises a circular cylindrical tube enclosing a water column (length H, cross-sectional area S f , mass M f = ρ f S f H ) supported at one end by a water-tight piston of mass M s , which is mounted on an elastic support of stiffness coefficient K s . The upper end of the water column is limited by a free surface at constant atmospheric pressure. As gravity effect on the

52


fluctuating pressure is neglected, the free surface condition is simply p ( H ) = 0 . The piston is assumed to slide freely along the tube axis and the tube is assumed to be rigid. Since the fluid is modelled as inviscid (non viscous) and incompressible, the water column has the same uniform displacement Z s as the piston. Therefore, the natural frequency of the system is immediately derived as: f =

1 2π

Ks 1 = f0 1+ μ f Ms + M f

[2.12]

f 0 is the natural frequency of the mass-spring system in vacuum and μ f is the

added mass ratio μ f = M f / M s . The result [2.12] can be established in a more formal way by using the Lagrange equations or the Rayleigh ratio (see AXI [04, 05]). Indeed the kinetic energy Eκ and the potential energy Ep of the system can be immediately expressed as: Ep =

1 K S Z s2 2

; Eκ =

1 M s + M f ) Z s2 ( 2

leading to the Rayleigh ratio: ω2 =

K s Z s2 Ks = 2 M ( M s + M f ) Zs s +Mf

It is noticed that to solve the present problem, there is no need to determine the pressure field induced by the vibration of the piston. This is because the kinetic energy of the fluid can be expressed directly in terms of the piston velocity. As further illustrated in a few following examples, such a shortcut is possible if, and only if, the fluid velocity can be inferred directly from the law of incompressibility. In all the other cases, the calculation of the pressure field cannot be avoided. Because of its extreme simplicity, the present example is convenient for introducing the method in a straightforward manner. 2.2.2.2 Piston-fluid system as a dynamically coupled system The static response of the piston to the weight of the water column is obtained by solving the equilibrium equation: K s Z 0 = −(Ps − Pa )S f

[2.13]

Z 0 is the displacement measured from the initial position of the unloaded piston, Ps

is the pressure exerted on the piston by the fluid, and Pa the atmospheric pressure, supposed to be constant. The later is governed by the hydrostatic equation:

Inertial coupling

∂P ∂z

= − ρ f g ⇒ P ( z ) = ρ f g ( H − z ) + Pa ; Z 0 = − gM f /K s

53

[2.14]

z =0

This elementary calculation suffices to point out that, in statics, the problem is uncoupled, because Ps = ρ f gH can be determined by using the fluid equation solely. Substituting the result in the structure equation [2.13], Z 0 = − gM f /K s is obtained as the solution of a forced problem. Such a conclusion remains the same in any other linear fluid-structure system. Turning now to the dynamic problem, described by using the equilibrium position of the piston loaded by the fluid weight as the state of reference, the harmonic vibration of the system is governed by a set of two coupled equations of the general type [2.4], which particularize here into: 1. Structure (mass-spring system):

( K δ ( z) − ω M δ ( z)) Z 2

s

s

s

+ p(z; ω )S f δ ( z ) = 0

[2.15]

In this particular example, the solid is a discrete system whereas the fluid is a continuum. Hence, to unify the mathematical treatment of the solid and the fluid parts of the problem, it is appropriate to multiply the coefficients of the mass-spring system by the Dirac distribution δ ( z ) ; which means that the stiffness and mass coefficients are viewed as operators concentrated at z = 0 . 2. Fluid column: 2

∂ p − ω 2 ρ f Z sδ ( z ) = 0 ∂ z2

[2.16]

The condition at the free surface is: p z=H = 0

[2.17]

To solve the system [2.15] to [2.17], we start by interpreting equations [2.15] and [2.16] in terms of action. For that purpose, they are integrated with respect to z on a small interval [ −ε , +ε ] , as already explained in [AXI 05], Chapter 3. Integration of [2.15] yields the discrete oscillator equation loaded by the pressure force:

(K

s

− ω 2 M s ) Z s + p(0; ω )S f = 0

[2.18]

Integration of the fluid equation [2.16] allows one to relate the fluctuating pressure field to the acceleration of the structure:

54


+ε

⌠ ⎮ ⎮ ⎮ ⎮ ⌡− ε

⎛∂ 2p ⎞ ∂p 2 ⎜ 2 − ω ρ f Z sδ ( z ) ⎟ dz = 0 ⇒ ∂z ⎝∂ z ⎠

− z =+ ε

∂p ∂z

= z =− ε

∂p ∂z

= ω 2 ρ f Zs

[2.19]

z =+ ε

The condition at the fluid-structure interface is deduced from [2.19] by letting ε tend to zero. Thus the equation [2.16] written in terms of distributions is found to be equivalent to the boundary value problem expressed in terms of ordinary functions: ∂ 2p =0 ∂ z2 ∂p ∂z

[2.20] 2

= ω ρ f Zs z =0

Solving [2.20] in conjunction with the condition [2.17] is straightforward. The result is: p(z; ω ) = −ω 2 ρ f ( H − z)Z s

[2.21]

By substituting the value of the pressure at the fluid-structure interface, as given by equation [2.21], into the structure equation [2.18], the equation of motion of an autonomous and harmonic oscillator is obtained, as should be. It reads as: K s Z s − ω 2 M s Z s − ω 2 ρ f S f HZ s = 0 ⇔

(K

s

−ω2 (Ms + M f

)) Z

s

=0

[2.22]

The method illustrated just above calls for the following comments: 1. Concerning the pair of equations [2.15] and [2.16], the use of the Dirac distribution to express the fluid-structure coupling terms arising at the vibrating wall allows us to formulate the problem in a condensed and rather symmetric manner. Indeed, in agreement with the general scenario of coupling described in Chapter 1, it is found that the fluid loads the structure by the wall pressure field, while the structure loads the fluid by a prescribed motion of the wall. To be more specific, the source term appearing in the fluid equation [2.16] may be interpreted as arising from the time derivative of a mass-flux per unit area of the wall. The presence of a time derivative is in agreement with the inertia principle of Galileo, according to which a constant mass-flux, or velocity, cannot induce any force or pressure. Finally, it is natural to put such source terms into the left-hand side of the equations because they stand for internal terms of the fluid-structure coupled system. 2. The calculation procedure follows the Newtonian approach as the pressure field is used to determine explicitly the force exerted on the structure by the fluid. 3. Once more, pressure stands for an intermediate variable but not for a degree of freedom of the coupled system. It is readily eliminated by using [2.21] to produce the canonical equation governing the free and harmonic vibrations of any conservative linear oscillator.

Inertial coupling

55

2.2.2.3 Piston-fluid system: tube of variable cross-section

Figure 2.5. Piston-fluid system: assembly of two distinct tubes

The system studied here differs from the former by the geometry of the water column which comprises two lengths of pipe of distinct cross-sections, see Figure 2.5 which specifies the geometric parameters of the problem. Adopting again the one-dimensional fluid column model, due to mass conservation, the fluid velocity field is readily found to be: S ; Z 2 = 1 Z s S2

Z1 = Z s

[2.23]

The kinetic energy of the fluid follows immediately as: H

⌠

1⌠ 1 1⎮ Eκ = ⎮⎮ ρ f S1Z s2 dz + ⎮⎮ 2 ⌡0 2⎮

H2

⎮ ⌡0

2

⎛S ⎞ M ρ f S2 ⎜ 1 ⎟ Z s2 dz = a Z s2 2 ⎝ S2 ⎠

[2.24]

The fluid added mass coefficient is: ⎛ ⎞ S M a = ρ f S1 ⎜ H1 + 1 H 2 ⎟ S2 ⎝ ⎠

[2.25]

The result [2.25] shows that the added mass can be smaller, or greater, than the actual fluid mass present in the system, depending whether the confinement ratio σ = S1 / S2 is smaller, or larger than unity. As a particular case, if S2 tends to zero, M a is found to tend to infinity, which means physically that the piston is blocked. Such a result could be anticipated since any volume change is impossible if the fluid is supposed to be incompressible. The importance of fluid inertia is suitably

56


measured by the added mass ratio μ f = M a / M S , which represents in fact the ratio of the amount of kinetic energy of the fluid to that of the solid. As already outlined in Chapter 1, subsection 1.3.3.1 and illustrated by the present exercise, if σ is finite and larger than one, the velocity of the upper confined fluid is larger than that of the vibrating structure and correlatively μ f is increased. On the other hand, if σ is less than unity, the velocity of the less confined fluid is smaller than that of the structure and μ f is decreased. Clearly, the same result is arrived at by using the Newtonian approach, presented here as an exercise to put in evidence a few interesting points concerning the fluctuating pressure field. The coupled problem is written as:

( K δ (z ) − ω M δ (z) ) Z s

S1

2

s

s

+ p(z; ω )S1δ (z ) = 0

[2.26]

∂ 2 p1 − ρ f S1ω 2 Z sδ (z) = 0 ; z ∈ [0,H1 ] 2 ∂z

[2.27]

∂ 2 p2 = 0 ; z ∈ [ H1 ,H1 + H 2 ] S2 ∂ z2

At the interface between the two fluid columns, the conditions [2.8] hold once more, because no mass is lost nor gained and no force is exerted there. Thus, the conditions of connection are written as: q1 (H1 ;ω ) = q2 (H1 ;ω ) ; p1 (H1 ;ω ) = p2 (H1 ;ω )

[2.28]

Once more, the q can be eliminated by using the momentum equation to get: iωρ f q1 (H1 ;ω ) = − S1

∂ p1 ∂z

; iωρ f q2 (H1 ;ω ) = − S2 z = H1

∂ p2 ∂z

[2.29] z = H1

Substituting [2.29] into [2.28], the continuity at H1 of mass flow is written as: ⎛ ∂ p2 ∂ p1 ⎞ − S1 =0 ⎜ S2 ⎟ ∂z ∂ z ⎠ z=H ⎝ 1

[2.30]

Solving the equations [2.27] which comply with the appropriate continuity and boundary conditions presents no particular difficulty. After a few manipulations the following results are obtained:

Inertial coupling

57

q(z; ω ) = iω S1Z s ⎛ ⎞ S p1 (z; ω ) = −ω 2 ρ f ⎜ H1 + 1 H 2 − z ⎟ Z s S2 ⎝ ⎠

[2.31]

⎛S ⎞ p2 (z; ω ) = −ω 2 ρ f ⎜ 1 ⎟ ( H − z ) Z s ⎝ S2 ⎠

The pressure field [2.31], normalized by the scale factor pr = −ω 2 ρ f HZ s , is shown in Figure 2.6 for three values of the confinement ratio. At H1 , the slope of the pressure profiles is discontinuous, in agreement with the condition [2.30].

Figure 2.6. Profile of the oscillating pressure along the pipe

2.2.2.4 Hole and inertial impedance

Figure 2.7. Tube ended by a rigid top provided with an aperture; equivalent 1D model

58


Let us consider the piston-fluid system sketched in Figure 2.7. Its upper end is bounded by a rigid wall provided with a circular hole, opening up on the free atmosphere. The hole area s f is varied from zero to the full cross-sectional area of the tube. If s f is equal to zero, M a is infinite, as already shown. On the other hand, if s f is equal to S f , the added mass is equal to the fluid mass M f contained in the tube, provided a pressure node is assumed at the aperture. The problem now is to determine the added mass coefficient M a in an intermediate case 0 < s f < S f . A priori, we could suggest the prescription of a node of mass-flux at any point of the top wall and a node of pressure at any point of the hole. However, such conditions are clearly incompatible with the 1D fluid column model. On the other hand, it is rather obvious that some amount of fluid oscillates back and forth from the aperture in the axial and the radial directions, invalidating the 1D model, at least locally. In fact, as the fluid is incompressible, its mean velocity through the hole is: S Z f ( H ) = f Z s sf

[2.32]

An oscillating pressure field must be related to such an oscillating flow, which necessarily varies in both the radial and axial directions. In particular, pressure cannot be strictly zero at H, in contradiction to the assumption made in the onedimensional model. Thus, in the vicinity of the hole inside and outside of the tube, the fluid motion is at least two-dimensional. An approximate and convenient way to deal with such a problem is to assume that the one-dimensional flow [2.32] is maintained practically unchanged over a characteristic length h related to the radius of the aperture. Accordingly, we are led to model the aperture as a virtual tube of length h and cross-sectional area s f . The auxiliary tube is opening up on the free atmosphere and the added mass is given by equation [2.25], rewritten here as: ⎛ ⎛S M a = M f ⎜1 + ⎜ f ⎜ ⎜ sf ⎝ ⎝

⎞⎛ h ⎞⎞ ⎟⎟ ⎜ ⎟ ⎟⎟ ⎠⎝ H ⎠⎠

[2.33]

The added mass coefficient given by [2.33] differs from that derived in subsection 2.2.2.2 even if S f = s f , because it includes a corrective term accounting for the oscillation of a certain mass of fluid outside the hole, which is discarded if a pressure node is assumed at H instead of at H + h . The appropriate value of the corrective length h can be calculated analytically from a 3D analysis, as will be shown in Chapter 7, subsection 7.2.1.4. In the case of a circular hole of diameter d, h is found to be: h=

4 d 3π

[2.34]

Inertial coupling

59

Substituting the value [2.34] into [2.33], the following dimensionless added mass coefficient is obtained: Ma 4 ⎛ D ⎞⎛ D ⎞ = μ f = 1+ ⎜ ⎟⎜ ⎟ 3π ⎝ d ⎠ ⎝ H ⎠ Mf

[2.35]

On the other hand, applying the local equations [2.10] to the virtual tube, for a harmonic oscillation we arrive at: q = iωρ f S f Z s −ω 2 ρ f S f Z s + s f

dp =0 dz

[2.36]

Due to fluid incompressibility, the pressure gradient in the virtual tube is: p(H ) dp =− dz h

[2.37]

Thus it is realized that we can arrive at the same result [2.33] by suitably modifying the boundary condition at the hole, instead of actually adding a virtual tube as we did just above. Relations [2.36] and [2.37] can be used to define the inertial impedance to be applied at the end of the main tube as: Z (H ) =

p(H ) p ( H ) iω h ρ f = = q ( H ) iω S f Z s sf

[2.38]

As we shall see later in Chapter 4, in acoustics an impedance is defined as the ratio of pressure to fluid particle velocity. However, as already indicated in Chapter 1, subsection 1.3.2.3, a mechanical impedance can also be defined in a generalized manner as the ratio of a stress variable over the dual kinematical variable, independently of their physical nature. For instance, in tubular piping systems, it is found convenient to choose the fluctuating volume-flow rate as the pertinent kinematical variable, see for instance [BLA 00]. 2.2.2.5 Response to a seismic excitation In agreement with the considerations made in [AXI 04], by seismic excitation we mean that the motion of some degrees of freedom or material points of the mechanical system is prescribed as given time functions of either displacement or acceleration. Such exciting fields are defined in an inertial frame. However, it is often preferred to describe the motion in a so-called relative frame moving according to the prescribed motion. Such kinds of excitation present several particularities of practical interest which are addressed here regarding a few arrangements of the piston-fluid column system.

60


Figure 2.8. Seismic excitation of the piston-fluid column system

In the simplest arrangement, the basement of the spring is excited by a prescribed transient displacement and the tube outlet is open. Denoting D ( t ) the prescribed displacement, X S the displacement of the piston as defined in the accelerated frame and YS that defined in the inertial frame, the coupled system is described by the coupled equations, written, for sake of brevity, in terms of distributions as: K sδ ( x ) X s + M sδ ( x ) Ys + pS f δ ( x ) = 0 ∂2 p + ρ f Ysδ ( x ) = 0 ∂ x2

;

p (H;t) = 0

[2.39]

Solving the fluid equation is immediate, giving the pressure field: p ( x; t ) = ρ f Ys ( H − x )

[2.40]

Substituting [2.40] into [2.39] leads to the oscillator equation: ( t ) K s X s + ( M s + M f ) Ys = 0 ⇔ K s X s + ( M s + M f ) X s = − ( M s + M f ) D

[2.41]

Equation [2.41] is the same as that which would be obtained by assuming that the forcing motion is prescribed to the structure and to the fluid as well. Such a result could be anticipated since in the present geometry the motion imparted to the structure is transmitted to the fluid without any change. However, it can be noted that if the tube outlet is closed, no relative motion is allowed between the piston and the tube. Of course, if the tube is rigid and fixed in the inertial frame, the system is blocked. If the tube is anchored to the moving ground the fluid-structure system remains at rest in the relative frame. This means that no net force is exerted on the coupled system in the relative frame. It is of interest to examine the problem using the mathematical formalism of the coupled equations. The system [2.39] becomes:

Inertial coupling

61

K sδ ( x ) X s + M sδ ( x ) Ys + pS f (δ ( x ) − δ ( H − x ) ) = 0 ∂2 p δ ( H − x ) = 0 + ρ f Ysδ ( x ) + ρ f D ∂ x2

[2.42]

Concentrating first on the fluid equation, it is easily checked that the only ⇔ X = 0 and the pressure field is found to be: possible solution is Ys = D s + p p ( x; t ) = − ρ f Dx 0

[2.43]

where p0 is an arbitrary constant. Substituting [2.43] into solid equation [2.42], there is a force unbalance unless the support reaction R of the tube on the ground is included in the balance to give the expected reaction force: R = −(Ms + M f ) D

[2.44]

Considering now the arrangement of Figure 2.5, if the tube is assumed to be fixed in the inertial frame the coupled equations are: K sδ ( x ) X s + M sδ ( x ) Ys + pS1δ ( x ) = 0 ∂ ⎛ ∂p ⎞ ⎜Sf ⎟ + ρ f Ys S1δ ( x ) = 0 ; ∂x ⎝ ∂ x ⎠

p ( H1 + H 2 ; t ) = 0

[2.45]

It is left to the reader, as a short exercise, to show that the oscillator equation is: K s X s + ( M s + M a ) Ys = 0 ⇔ K s X s + ( M s + M a ) X s = − ( M s + M a ) D

[2.46]

where the added mass coefficient is given again by formula [2.25]. On the other hand, if the tube is anchored to the moving ground as suggested in Figure 2.5, the coupled equations become: K sδ ( x ) X s + M sδ ( x ) Ys + pS1δ ( x ) = 0 ∂ ⎛ ∂p ⎞ ⎜Sf ⎟ + ρ f Ys S1δ ( x ) + ρ f D ( S2 − S1 ) δ ( H1 − x ) = 0 ∂x ⎝ ∂ x ⎠ p ( H1 + H 2 ; t ) = 0

[2.47]

The pressure field varies linearly along each tube segment as: p1 = − ρ f Ys x + p0

; 0 ≤ x ≤ H1

p2 = a ( x − H1 − H 2 )

; H1 ≤ x ≤ H 2

[2.48]

62


The constants p0 and a, are determined by using the connecting conditions at the abrupt change in the cross-sectional area, namely the continuity pressure at H1 and the finite jump of the volume velocity due to the prescribed motion of the fluid at H1 . Hence p0 and a are governed by the two equations: p0 = ρ f Ys H1 − aH 2

[2.49]

aS2 + ρ f Ys S1 = ρ f ( S2 − S1 ) D

Solution is immediate and the pressure exerted on the piston is found to be: ⎛ ⎞ S p0 = ρ f ⎜ H1 + 1 H 2 ⎟ Ys − ρ f S 2 ⎝ ⎠

⎛ S1 ⎞ ⎜1 − ⎟ H 2 D ⎝ S2 ⎠

[2.50]

Substituting pressure [2.50] into the first equation [2.47] produces the forced oscillator equation written in the relative frame as: K s X s + ( M s + M a ) X s = ( M t − ( M s + M a ) ) D

[2.51]

Presence of the incompressible fluid is now characterized by two distinct mass coefficients, namely the added mass coefficient M a which again is given by formula [2.25], and the so called transported mass coefficient: M t = ρ f ( S2 − S1 )

S1 H2 S2

[2.52]

Depending whether S2 is larger, or at the opposite smaller than S1 , the seismic loading is less, or larger than that predicted by the added mass component. Finally, to conclude this subsection it is of interest to analyse the seismic response of this kind of system, as function of two distinct dimensionless parameters, namely, the ratio τ 0 = T0 / Te of the characteristic time of the excitation on the period of the oscillator in vacuum and the added mass ratio μ f = M a / M s . In the case of a non uniform tube, an additional pertinent parameter μt = M t / M a would to be considered also. However, for the sake of brevity, the problem is restricted here to the case of a tube of constant cross-section. On the other hand the prescribed displacement is selected as the following transient:

Inertial coupling

D ( t ) = D0 sin ωe t ( U (t ) − U ( t − Te ) )

63

[2.53]

U ( t ) designates the Heaviside step function, ωe is the circular frequency and Te

the period of the exciting sine function, see the upper plot of Figure 2.9.

Figure 2.9. Seismic excitation signal and response of a mass-spring system displayed as the maximum displacement of the oscillator versus the frequency ratio f r of the exciting truncated sinus to the natural frequency of the oscillator

Substituting the displacement [2.53] into equation [2.51], the Laplace transform of X s is found to be (on the use of the Laplace transform, see for instance [AXI 04]):

64


X s ( s ) =

ωe3 D0 (1 − e −Te s )

(ω

2 e

+ s 2 )(ω02 + s 2 )

[2.54]

It is appropriate to distinguish the response during the forced stage ( t ≤ Te ) and that during the free stage of motion. The inverse Laplace transform of [2.54] is found to be: ⎧ X s sin ω0 t − ϖ 0 sin ωe t t ≤ Te 2 ⎪ D = 1 ϖ ϖ − ( ) 0 0 0 ⎪ ⎨ ⎪ X s = sin ω0t − sin ω0 ( t − Te ) t > T e ⎪ D0 ϖ 0 (ϖ 02 − 1) ⎩

[2.55]

The frequency ratio ϖ 0 = ω0 / ωe = 1/ f r is used as a pertinent dimensionless parameter. If ϖ 0 = 1 , the solution [2.55] becomes: ω0t cos ω0t + sin ω0t ⎧ Xs t ≤ Te T0 ⎪D − 2 ⎪ 0 ⎨ Xs ⎪ = π cos ω0t t > Te ⎪⎩ D0

[2.56]

Of course, in the present analysis, it is tacitly assumed that compressibility of the fluid can be neglected whatever the value of the characteristic times T0 and Te may be. Effect of fluid compressibility shall be addressed later, in Chapter 6, subsection 6.2.3.2. The excitation signal is shown in the upper plot of Figure 2.9, where the reduced time is defined as tr = t / T0 . In the lower plot of Figure 2.9 the maximum dimensionless displacement max X s ( t ) / D0

of the oscillator is plotted as a

function of f r . Since seismic excitation means force of inertia, the magnitude of the response increases with the excitation frequency. However, in the high frequency range, it tends asymptotically to a finite value, namely max( X max / D0 ) = 2π , because action of the force remains finite, as illustrated in [AXI 04] taking the example of a car crossing a bump. On the other hand, the curve is independent on the effective mass M s + M a which is a mere consequence of the fact that the same mass coefficient appears in the right and in the left hand-side of the equation of motion [2.41].

Inertial coupling

65

Figure 2.10. Maximum displacement of the mass-spring system to a given seismic signal plotted versus the reduced added mass coefficient μ f

Figure 2.11. Transition between two response signatures of the mass spring-system leading to an abrupt change in the slope of the curve of the maximum displacements at f e / f 0 = 0.5

However, it is important to emphasize that the lower plot of Figure 2.9 can be interpreted differently, by considering a given oscillator coupled to an incompressible fluid and excited by a given seismic signal. As the natural frequency of the oscillator is sensitive to the added mass coefficient, so is the seismic response; which emphasizes the relevance of μ f as an important parameter of the problem. Dependence may be very significant, or not, depending whether f r is less than

66


unity or not, as illustrated by the two plots of Figure 2.10. Incidentally, a few kinks are present in the curves of Figure 2.9 at particular frequencies, which are conspicuous in the upper left plots of Figures 2.11 and 2.12. As indicated in the time histories displayed in the other plots, they correspond to a transition between slightly distinct features of the response signal which is marked by the disappearance of the free oscillations (cf. [AXI 04], Chapter 7).

Figure 2.12. Transition between two response signatures of the mass spring-system leading to an abrupt change in the slope of the curve of the maximum displacements at f e / f 0 = 0.2

2.2.2.6 Nonlinear inertia in piping systems

Figure 2.13. Piping system including a conical tube

Inertial coupling

67

In tubes of uniform cross-section, the convective inertia present in the nonlinear momentum equation [1.43] vanishes whatever the magnitude of the oscillation may be. This is no longer the case when the pipe cross-section varies because the axial component of the fluid velocity is inversely proportional to the cross-sectional area, in order to satisfy the mass equation. Hence, it is of interest to investigate the consequence of the convective inertia on the oscillation of the piston. For that purpose, let us consider the system sketched in Figure 2.13. The pipe comprises three distinct segments. The first one has a length L1 and a constant cross-sectional area S1 . It is connected to a conical tube of length L2 and cross-sectional area

S f ( x ) = S1 + ( S2 − S1 )( x − L1 ) / L2 . The third tube has a length L3 and a constant cross-sectional area S2 . The axial fluid velocity X f ( x; t ) is readily found to be:

⎧ ⎪ ⎪ ⎪⎪ X f ( x; t ) = ⎨ ⎪ ⎪ ⎪ ⎪⎩

X s

0 ≤ x ≤ L1 S1 X s L1 ≤ x ≤ L1 + L2 S f ( x) S X s 1 S2

[2.58]

L1 + L2 ≤ x ≤ L1 + L2 + L3

The kinetic energy of the fluid is calculated in the actual configuration of the column as: L +L ⌠ 1 2 ⎛ ⎞ 2 ⎮ ⎛ S1 ⎞ ⎛ S1 S1 ⎞ ⎟ 1 ⎜ 2 ⎮ Eκ ( X s , X s ; t ) = ρ f X s ⎜ ( L1 − X s ) S1 + S1 ⎮ dx + ⎜ ⎟ ⎜ L3 + X s ⎟ S2 ⎟ S f ( x) S2 ⎠ ⎟ 2 ⎝ S2 ⎠ ⎝ ⎮ ⎜ ⌡ L1 ⎝ ⎠

which is conveniently rewritten as: Eκ ( X s , X s ; t ) =

⎛ ⎛ ⎛ S ⎞2 ⎞ ⎞ 1 ρ f X s2 S1 ⎜ L − X s ⎜ 1 − ⎜ 1 ⎟ ⎟ ⎟ ⎜ ⎝ S2 ⎠ ⎟ ⎟ ⎜ 2 ⎝ ⎠⎠ ⎝

[2.59]

where an equivalent length of the piping is defined as: L + L2

L=

⌠ 1 ⎮ L1 + ⎮⎮ ⎮ ⌡L1

⎛S ⎞ dx + L3 = L1 + L2 + L3 ⎜ 1 ⎟ S f ( x) ⎝ S2 ⎠ S1

The equivalent length of the conical tube is found to be:

[2.60]

68

Fluid-structure interaction L + L2

L2 =

⌠ 1 ⎮ ⎮ ⎮ ⎮ ⌡L1

S1

S f ( x)

dx = L2

⎛ ⎞ S1 L2 S2 ln ⎜⎜ ⎟ S2 − S1 ⎝ L2 S1 + L1 ( S2 − S1 ) ⎟⎠

[2.61]

The force exerted on the piston follows as: ⎛ d ⎛ ∂E ⎞ ∂E FI = − ⎜⎜ ⎜ κ ⎟ − κ ⎝ dt ⎝ ∂X S ⎠ ∂X S

⎞ ⎛ X s2 ⎞ β − X s ( L − X S β ) ⎟ ⎟⎟ = ρ f S1 ⎜ ⎝ 2 ⎠ ⎠

2

⎛S ⎞ where β = 1 − ⎜ 1 ⎟ . ⎝ S2 ⎠

[2.62]

The result [2.62] shows that nonlinear effects cancel if S1 = S2 and this,

independently of the intermediate values of S f ( x ) . This indicates that any excess

of convective inertia in those parts of the piping where the cross-section diminishes is exactly balanced by a default of inertia in those parts where the cross-section increases. Thus if S1 = S2 , the sole effect on the added mass induced by the crosssectional variations of the piping is related to the change in its equivalent length. As expected, L2 is shorter or bigger than L2 , depending on whether S2 is larger or smaller than S1 . Correlatively, the added mass is smaller, or larger than its value in a tube of uniform cross-section S1 and total length L = L1 + L2 + L3 . Furthermore, even if β differs from zero, the nonlinear inertia remains negligible provided X s β PL and to initial gas pressure small enough so that water velocity remains substantially smaller than the speed of sound in water. In addition, dissipative effects such as viscous friction and energy loss by radiation of the sound waves triggered by the explosion are also neglected. Since ρ L >> ρG , contribution of the gas to the kinetic energy of the system is negligible. Potential energy related to the gas expansion is readily found to be: EG =

4π PG 0 R 3 ⎛ R0 ⎞ ⎜ ⎟ 3 (γ − 1) ⎝ R ⎠

3γ

[2.75]

Figure 2.15. Potential energy related to the bubble expansion

Potential energy related to the liquid is: EL =

4π PL R 3 3

[2.76]

In Figure 2.15, the potential energy of a bubble is plotted for a few values of PL , assuming PG 0 = 100 Mpa at R0 = 0.5m . As could be anticipated, the potential is marked by a minimum at some equilibrium radius Re . Of course, Re is controlled by the relative values of PG 0 and PL , tending to infinity if PL tends to zero. In the range R > Re , the liquid term prevails leading to a positive and more gradual slope of the potential. Since we deal with a conservative and single degree of freedom system, the expansion law of the bubble can be calculated semi-analytically based on the invariance of mechanical energy (cf. [AXI 04], Chapter 5). Starting from an initial state at rest, the energy balance is written as : 3γ ⎛1 ⎛ cb2 cb2 ⎛ R0 ⎞ cL2 ⎞ cL2 ⎞ 3 Em = 4πρ L R 3 ⎜ R 2 + + = + 4 R πρ ⎟ ⎜ ⎟⎟ L 0 ⎜ ⎜ ⎟ ⎜2 3 (γ − 1) ⎝ R ⎠ 3 ⎟⎠ ⎝ 3 (γ − 1) 3 ⎠ ⎝

[2.77]

where cb and cL are defined as the following characteristic speeds: cb =

PG 0 ρL

PL = α cb ρL

; cL =

; α=

PL 1, in which case it turns out that the boundary conditions at z = 0 and z = H can be discarded, as a first approximation at least. Such a simplifying assumption corresponds to the strip model broadly used in engineering to compute the added mass coefficients of slender elongated bodies immersed in, or containing, a fluid, see for instance [FRI 72], [SIG 03]. The basic assumption of the strip model from the fluid standpoint is that inside a narrow strip between z and z+dz, located sufficiently far from the ends z = 0 and z = H, the axial flow component is negligible, see Figure 2.22. Furthermore, it is also assumed that

86


the end effects extend over a small axial length only, in such a way that their relative importance is small on the scale of the whole system. Validity of such assumptions will be discussed in subsections 2.3.5.

Figure 2.22. Strip of a circular cylindrical shell

From the structural standpoint, when writing the shell equations the axial component of motion is discarded and so are the axial variations of the displacement field. Therefore, the radial and tangential Love equations are written as: Es e 1 −ν s2

⎧⎪ U e2 ⎨ 2+ 2 ⎩⎪ R 12 R

⎛ ∂ 4U ∂ 3V ⎞ ∂V ⎫⎪ ⎜ 2 4 − 2 3 ⎟ + 2 ⎬ + ρ s eU = p ( R, θ ; t ) R ∂θ ⎠ R ∂θ ⎭⎪ ⎝ R ∂θ

Es e ⎧⎪⎛ e2 ⎞ ⎛ ∂ 2V ⎞ ∂U e 2 ∂ 3U ⎫⎪ 1 + + − ⎨ ⎬ − ρ s eV = 0 ⎜ ⎟ ⎜ ⎟ 1 −ν s2 ⎪⎩⎝ 12 R 2 ⎠ ⎝ R 2 ∂θ 2 ⎠ R 2 ∂θ 12 R 2 R 2 ∂θ 3 ⎭⎪

[2.123]

As could be expected a priori, the system [2.123] is nearly the same as that which governs the in-plane modes of a circular ring coupling bending and axial vibration (cf. [AXI 05], Chapter 8). Furthermore, it was shown that coupling between inplane bending and axial vibrations can be neglected as a first approximation, leading to the pure bending model according to which the hoop strain is assumed to vanish: ηθθ =

U ∂V + =0 R R∂θ

[2.124]

As a consequence, to alleviate the calculation, the system [2.123] is replaced by the following radial equation: Es e 3

⎛ ∂ 4U ∂ 2U ⎞ + ⎜ ⎟ + ρ s eU = p ( R,θ ; t ) 4 ∂θ 2 ⎠ 12 (1 −ν s2 ) R 4 ⎝ ∂θ

[2.125]

Inertial coupling

87

The mode shapes are of the following admissible type: un (θ ) = α n cos nθ + β n sin nθ ⎫ ⎬ n = 1, 2,... vn (θ ) = an cos nθ + bn sin nθ ⎭

[2.126]

which can be conveniently split into two orthogonal families of mode shapes: 1 1 1 un( ) (θ ) = cos nθ ; vn( ) (θ ) − sin nθ n 1 ( 2) (2) un (θ ) = sin nθ ; vn (θ ) cos nθ n

[2.127]

Figure 2.23. In-plane translation modes of the cosine and sine families

The corresponding stiffness and mass coefficients per unit shell length are given by: 2π

ms(

1,2 )

k s(

1,2 )

( n, n ) = ( n, n ) =

⌠ ⎮ ρ s e ⎮⎮ ⎮ ⎮ ⌡0

Es e

1 ⎞ ⎧cos2 nθ ⎫ 1 ⎞ ⎛ ⎛ ⎜ 1 + 2 ⎟ ⎨ 2 ⎬ R d θ = eπ R ⎜ 1 + 2 ⎟ n n sin n θ ⎝ ⎠⎩ ⎝ ⎠ ⎭ 2π

⌠ n n −1 ⎮ ⎮ 1 − ν s2 R 3 ⎮⎮⎮ ⌡0

3 2

12 (

(

2

)

)

Es e π n ( n − 1) ⎧cos nθ ⎫ ⎨ 2 ⎬ R dθ = 12 (1 − ν s2 ) R 3 ⎩ sin nθ ⎭ 3

2

2

[2.128]

2

The natural circular frequencies in vacuum are: Ωn =

k s ( n, n ) n 2 ⎛ e ⎞ Es = ⎜ ⎟ ms ( n, n ) R ⎝ R ⎠ 12 (1 −ν s2 ) ρ s

⎛ n2 − 1 ⎞ ⎜ 2 ⎟ ⎝ n +1⎠

[2.129]

88


Free and rigid in-plane translations correspond to n = 1 , see Figure 2.23. The mode shapes constitute a subspace spanned by the two orthonormal vectors: u1( ) = cos θ

; v1( ) = − sin θ ;

1

1

u1( ) = sin θ 2

; v1( ) = cos θ 2

[2.130]

The modes n > 1 correspond to axial bending see Figure 2.24.

Figure 2.24. Bending mode shapes of the cosine family

We turn now to the motion of the fluid forced by a radial vibration of the shell of the type: U (θ ; t ) =

n =+∞

∑ ( q( ) cos nθ + q( ) sin nθ ) n =1

1 n

2 n

[2.131]

where the time functions qn(1) (t ) and qn( 2) (t ) stand for the modal displacements of the shell. Pressure is governed by the boundary value problem: ∂2p 1∂ p 1 ∂2p + + =0 ∂ r 2 r ∂ r r 2 ∂θ 2 ∂p ∂r

r=R

= −ρ f

n =+∞

∑ ( q n =1

(1)

n

( 2)

cos nθ + qn sin nθ

)

[2.132]

Inertial coupling

89

Following the mathematical procedure described in subsection 2.3.1, the problem [2.132] is replaced by the simpler problems: ∂ 2 pn(1) 1 ∂ pn(1) 1 ∂ 2 pn(1) + + 2 =0 r ∂r r ∂θ 2 ∂ r2 ∂ pn( ) ∂r 1

= − ρ f qn( ) cos nθ 1

r =R

[2.133]

∂ 2 pn( ) 1 ∂ pn( ) 1 ∂ 2 pn( ) + + 2 =0 r ∂r r ∂θ 2 ∂ r2 2

∂ pn( ) ∂r

2

2

r=R

2

= − ρ f qn( 2 ) sin nθ

As detailed just below, they can be solved by using the separation method. Assuming for both of them a solution of the type: pn(1,2) (r ,θ ; t ) = S n (r )Tn(1,2) (θ )bn(1,2) (t )

[2.134]

the partial derivative equation is suitably replaced by the following system of ordinary differential equations: r 2 ⎛ d 2 S n 1 dS n + ⎜ S n (r ) ⎝ dr 2 r dr

⎞ 2 ⎟ − kn = 0 ⎠

[2.135]

d 2Tn(1,2) + kn2Tn(1,2) (θ ) = 0 dθ 2

Since pressure is necessarily of period 2π  in θ, the constant kn must be positive, in such a way that Tn(1,2) (θ ) is of the sinusoidal type: Tn(1,2) (θ ) = An cos knθ + Bn sin knθ

[2.136]

The constants An , Bn are specified by using the appropriate conditions at the fluidstructure interface. It follows that: 1 1 1 bn( ) (t )Tn( ) (θ ) = − ρ f qn( ) cos nθ

2 2 2 ; bn( ) (t )Tn( ) (θ ) = − ρ f qn( ) sin nθ

⇒ kn = n ; bn(1) (t ) = − ρ f qn(1)

; bn( 2) (t ) = − ρ f qn( 2)

[2.137]

Substituting kn = n into the radial equation [2.135], the general solution is found to be of the type: Sn (r ) = α n r n + β n r - n

[2.138]

90


Once more α n and β n are constants to be adjusted to the particularities of the physical problem treated. In the present case, β n is necessarily nil, since the pressure cannot be infinite on the cylinder axis. On the other hand, α n is specified by substituting the physically admissible solution S n (r ) = α n r n into the condition to be fulfilled at the fluid-structure interface: ∂ pn( ) ∂r

⎫ 1 1 = −nα n R n-1 ρ f qn( ) cos nθ = − ρ f qn( ) cos nθ ⎪ 1 ⎪ ⎬ ⇒ α n = n-1 ( 2) nR ∂ pn ⎪ = −nα n R n-1 ρ f qn( 2 ) sin nθ = − ρ f qn( 2 ) sin nθ ⎪ r R = ∂r ⎭ 1

r =R

[2.139]

The pressure field induced by the motion of the shell according to the n-th modes of vibration [2.127] is thus: pn( ) (r ,θ ; t ) = − ρ f 1

1 qn( ) n r cos nθ nR n-1

;

pn( ) (r ,θ ; t ) = ρ f 2

2 qn( ) n r sin nθ nR n-1

[2.140]

As the structural and the fluid problems have the same axial symmetry, in what follows it will suffice to retain only one of the two mode families, for instance the cosine one. The equation [2.125] is thus written as: m =+∞ (1) ⎛ ∂ 4U ∂ 2U ⎞ = − ρ ∑ qm R cos mθ ρ + + eU ⎜ ⎟ s f 4 ∂θ 2 ⎠ m 12 (1 −ν s2 ) R 4 ⎝ ∂θ m =1

Es e 3

[2.141]

Modal projection of [2.141] is straightforward. The pressure fields [2.140] and the radial mode shapes [2.127] have the same θ profiles, which verify the orthogonality conditions: 2π

⌠ ⎮ ⌡0

⎧π cos nθ cos mθ d θ = ⎨ ⎩0

n=m≠0 n≠m

[2.142]

Therefore, in the present problem, the added mass matrix, as expressed in the structural modes basis, is diagonal and the mode shapes of the shell are the same as in vacuum φ n ≡ Φ n . The generalized force exerted by the fluid on a shell strip of unit length is: Qn = − ρ f

2π ρ f π R 2 (1) qn(1) 2 ⌠⎮ R ⎮ ( u.u ) cos 2 nθ d θ = − qn n n ⌡0

[2.143]

where u is the unit vector in the radial direction, see Figure 2.21. In agreement with the definition [2.113], the added mass coefficients per unit length of the shell are:

Inertial coupling

⌠

ma ( n, m ) = ⎮⎮

2π

⌡0

⎧mf ⎪ pn(1) ( R,θ )Φ m (θ ).n Rdθ = ⎨ n ⎪ 0 ⎩

n=m≠0

91

[2.144]

n≠m

where the mode shape is normalized by the condition max un(1,2) (θ ) = 1 and m f = ρ f π R 2 stands for the physical mass of the fluid contained in a shell strip of

unit length. These results call for the following comments: 1. The added mass of the breathing mode n = 0 is infinite. As already indicated in subsection 2.2.3.1, this result is natural, since any variation of volume is prevented in an incompressible fluid. 2. The added mass coefficient of the modes n = 1 is equal to the physical mass m f . This result also could be easily anticipated, since the fluid must follow the uniform translation of the shell strip. 3. The fact that the added mass coefficients are found to decrease as 1/n indicates that the fluid motion diminishes with the circumferential wave number of the oscillations. Since the fluid flows from a pressure crest toward a trough, fluid motion is directly related to the r n law of the wavy pressure, as made clear by calculating the kinetic energy. Fluid velocity X is given by the momentum f

equations in the radial and circumferential directions: n-1 n-1 ⎛r⎞ ⎛r⎞ ρ f X f .u − ρ f qn(1) ⎜ ⎟ cos nθ = 0 ⇒ X f .u = U f = qn(1) ⎜ ⎟ cos nθ ⎝R⎠ ⎝R⎠ [2.145] n-1 n-1 r r ⎛ ⎞ ⎛ ⎞ (1) (1) ρ f X f .u1 + ρ f qn ⎜ ⎟ sin nθ = 0 ⇒ X f .u1 = Vf = −qn ⎜ ⎟ sin nθ ⎝R⎠ ⎝R⎠ where u and u1 denote the radial and tangential unit vectors, see Figure 2.21. Relations [2.145] show that the amplitude of the fluid oscillation decreases as a power law of index n-1 when one moves from the shell toward the cylinder axis. The kinetic energy is found to be:

Eκ =

ρ f ( qn(1) ) 2

2 ⌠ 2π

⎮ ⎮ ⎮ ⎮ ⌡0

R

⌠ ⎮ ⎮ ⎮ ⎮ ⌡0

⎛r⎞ ⎜ ⎟ ⎝R⎠

2 ( n-1)

ρ f ( qn(1) ) π R 2 2

rdrdθ =

2n

[2.146]

The fluid oscillations of a few modes are shown in Figure 2.25, where the acceleration field of the fluid is visualized by arrows. The pressure field is visualized in colour plate 1, where pressure crests are in red and pressure troughs in blue.

92


Figure 2.25. Fluid oscillations related to the shell modes: the arrows correspond to the acceleration field and the full line to the isobars; see also colour plate 1

2.3.2.2 Cylindrical shell immersed in an infinite extent of liquid It turns out that the preceding results hold also for the external problem of a circular cylindrical shell immersed in an infinite extent of liquid. The only difference lies in the radial solution of the pressure field [2.138]. Here, the coefficients α n must be nil because the fluctuating pressure vanishes necessarily at infinity. The pressure at the fluid-structure interface is thus: pn ( R, θ ; t ) = ρ f

Rqn(1) cos nθ n

[2.147]

Inertial coupling

93

Figure 2.26. Fluid oscillations related to the shell modes; see also colour plate 2

The modal force exerted by the fluid on a shell strip of unit length is: Qn = − ρ f

2π m f (1) qn(1) 2 ⌠⎮ R ⎮ ( −u.u ) cos 2 nθ d θ = − qn n n ⌡0

[2.148]

The relative importance of the fluid to the shell inertia is given by the ratio of the generalized mass coefficients: μ f ( n, n ) =

ma ( n, n ) ms ( n, n )

=

ρ f π R2 n ρ sπ eR

=

ρf R nρs e

[2.149]

which is similar to the result [2.71] found for the breathing mode of a spherical shell, except for the coefficient 1/n due to the shell and fluid wavy motion. Finally, the fluid oscillations are visualized in Figure 2.26, see also colour plate 2. It may be

94


noticed that the fluid motion differs markedly with respect to the internal case, even if the kinetic energy is the same. 2.3.2.3 Inertial coupling of two coaxial circular cylindrical shells

Figure 2.27. Coaxial circular cylindrical shells coupled by a liquid

The system is sketched in Figure 2.27. As in the preceding problems and for the same reasons, coupling occurs between the modes of the same circumferential index solely. Here the admissible radial solution of the pressure field is the complete form [2.138]. The conditions at the fluid-structure interfaces for radial shell harmonic vibrations of magnitudes un (1) and un ( 2 ) respectively, are: ∂ pn ∂r

r = R1

= ω 2 ρ f un (1) cos nθ ;

∂ pn ∂r

r = R2

= ω 2 ρ f un ( 2 ) cos nθ

[2.150]

Referring to equation [2.138], the coefficients α n , β n are calculated by inverting the matrix equation: ⎡ R1( n −1) ⎢ ( n −1) ⎣⎢ R2

− R1 (

− n +1)

⎤ ⎡α n ⎤ ω 2 ρ f ⎥⎢ ⎥ = n − R2−( n +1) ⎦⎥ ⎣ β n ⎦

⎡ un (1) ⎤ ⎢ ⎥ ⎣un ( 2 ) ⎦

[2.151]

The result is: ω 2 ρ f ⎛ un ( 2 ) R2n+1 − un (1) R1n+1 ⎞ ⎜ ⎟⎟ n ⎜⎝ R22 n − R12 n ⎠ 2 n+1 2 n ω ρ f ⎛ un ( 2 ) R2 R1 − un (1) R1n+1 R22 n βn = ⎜ n ⎜⎝ R22 n − R12 n

αn =

⎞ ⎟⎟ ⎠

The wall values of the pressure functions Sn ( r ) are:

[2.152]

Inertial coupling

⎛ ⎞⎛ ⎛ ⎛ R ⎞n ⎛ R ⎞n ⎞ ⎞ R1n R2n ⎜ ⎟ 2 R u ( 2 ) − R1un (1) ⎜ ⎜ 1 ⎟ + ⎜ 2 ⎟ ⎟ ⎟ Sn (R1 ) = ω ρ f ⎜ ⎜ R ⎜ n ( R22 n − R12 n ) ⎟ ⎜ 2 n R ⎟⎟ ⎝⎝ 2 ⎠ ⎝ 1 ⎠ ⎠⎠ ⎝ ⎠⎝

95

2

⎛ ⎞⎛ ⎛ ⎛ R ⎞n ⎛ R ⎞n ⎞ ⎞ R1n R2n ⎜ ⎜ ⎟ −2 R1un (1) + R2un ( 2 ) ⎜ ⎜ 1 ⎟ + ⎜ 2 ⎟ ⎟ ⎟ Sn (R2 ) = ω ρ f ⎜ R ⎜ n ( R22 n − R12 n ) ⎟ ⎜ R ⎟⎟ ⎝⎝ 2 ⎠ ⎝ 1 ⎠ ⎠⎠ ⎝ ⎠⎝

[2.153]

2

The generalized forces exerted by the fluid on the shell strips per unit length are: 2π

⌠ Qn (1) = ( −u .u ) ⎮⎮ Sn ( R1 ) cos 2 nθ R1dθ ⌡0

[2.154]

2π

⌠ Qn ( 2 ) = ( +u.u ) ⎮⎮ Sn ( R2 ) cos 2 nθ R2 dθ ⌡0

Substituting the pressures [2.153] into [2.154] we arrive at: Qn (1) = Qn ( 2 ) =

ω2ρ f π

n(R

2n 2

2n 1

−R

ω2ρ f π

(R (R )

n ( R22 n − R12 n )

2 1

2n 1

(R (R 2 2

2n 1

+ R22 n ) un (1) − 2 ( R1 R2 ) 2n 2

+R

n+1

) u ( 2) − 2 ( R R ) n

1

un ( 2 )

n+1

2

)

un (1)

)

[2.155]

Whence the added mass coefficients per unit length of the shells: mn( a ) (1,1) = (a) n

m

ρ f π R12 ⎛ R12 n + R22 n ⎞ ⎜ ⎟; n ⎝ R22 n − R12 n ⎠

(1, 2 ) = m

(a) n

mn( a ) ( 2, 2 ) =

n 2 ρ f π R1 R2 ⎛⎜ ( R1 R2 ) ( 2,1) = − ⎜ R22 n − R12 n n ⎝

ρ f π R22 ⎛ R12 n + R22 n ⎞ ⎜ ⎟ n ⎝ R22 n − R12 n ⎠ ⎞ ⎟ ⎟ ⎠

[2.156]

The added mass matrix is positive definite, as easily checked by looking at the sign of the eigenvalues. The later are the roots of the algebraic equation: λ 2 − bλ + c = 0 b = mn( a ) (1,1) + mn( a ) (2, 2) =

ρ f π ( R22 + R12 ) ⎛ R12 n + R22 n ⎞ >0 ⎜ 2n 2n ⎟ n ⎝ R2 − R1 ⎠

c = m (1,1)m (2, 2) − ( m (1, 2) ) (a) n

(a) n

(a) n

2

⎛ ρ f π R2 R1 =⎜ ⎜ n ⎝

[2.157]

2

⎞ ⎟⎟ > 0 ⎠

which are found to be both positive, since the coefficients b and c are positive too.

96


On the other hand, the coupled vibration modes are the solution of the matrix equation: (S ) (a) ⎡ ⎡ kn( S ) (1,1) ⎤ ⎤ ⎤ ⎡ un (1) ⎤ ⎡0⎤ 0 mn( a ) (1, 2) 2 ⎡ mn (1,1) + mn (1,1) ⎢⎢ ⎥ = ⎢ ⎥ [2.158] ⎥ −ω ⎢ ⎥⎥ ⎢ (S ) (a) (S ) (a) 0 k (2, 2) m (1, 2) m (2, 2) m (2, 2) + n n n n ⎦ ⎣ ⎦ ⎦⎥ ⎣un ( 2 ) ⎦ ⎣0⎦ ⎣⎢ ⎣

The matrix equation [2.158] indicates that the modes of index n of each shell in vacuum give rise to two distinct modes of index n coupled by the fluid inertia. The product un (1) un ( 2 ) is positive for the in-phase mode and negative for the out-ofphase mode.

(a) uncoupled modes in vacuum

(b) modes coupled by the liquid Figure 2.28. Two concentric shells R1 / R2 = 0.5 : modes of vibration n = 3

Figure 2.28 shows the modes n = 3 of a pair of aluminium shells (thickness e = 1cm, inner and outer radii R1 = 0.5 m , R2 = 1m ). In vacuum, each shell vibrates independently from the other, the highest natural frequency corresponding to the mode of the smallest shell. If the annular space is filled with water, the modes are coupled by the fluid inertia and the natural frequencies are substantially less than in vacuum. As displayed in colour plates 1 and 2, the relative importance of fluid coupling decreases as n increases due to the radial variation of the fluctuating pressure. Furthermore, as can be anticipated, the frequency of the out-of-phase

Inertial coupling

97

mode is less than that of the in-phase mode, simply because the fluid is more squeezed, hence accelerated, when the shells vibrate out-of-phase than in-phase. Due to the rather large difference in the shell radii, the magnitude of the shell vibrations largely differs from one to the other, by a ratio which depends on the phase.

(a) uncoupled modes in vacuum

(b) modes coupled by the liquid Figure 2.29. Two concentric shells R1 / R2 = 0.8 : modes of vibration n = 3

Figure 2.29 refers to the same system except that R1 = 0.8 m , R2 = 1m . In vacuum the two shells vibrate independently at similar frequencies. In water, due to the small difference in the shell radii, the relative magnitude of the shell vibrations is nearly equal to one. So the squeezing of the fluid induced by the shell displacement according to the out-of-phase mode is much more pronounced than in Figure 2.28. Correlatively, the added mass coefficient of the in-phase mode is much smaller than that of the out-of-phase mode, as clearly indicated by the relative values of the natural frequencies.

98


2.3.3

Thin fluid layer approximation

2.3.3.1 Concentric cylindrical shells of revolution Returning to the system analysed in the last subsection, the case of a large confinement ratio σ f is of special interest, as it corresponds to a thin layer of fluid which can be treated in a simplified manner, as shown below. In the geometry considered here σ f is suitably defined as: σf =

R2 + R1 R >> 1 2 ( R2 − R1 ) h

[2.159]

where R2 R1 = R and R2 − R1 = h . As a first approximation, the radial profile of the fluctuating pressure in the annular space can be written as: pn ( r,θ ) =

⎛ ⎛ r ⎞ n ⎛ r ⎞- n ⎞ − u u 2 1 ( ) ( ) (n ) ⎜⎜ ⎜ R ⎟ + ⎜ R ⎟ ⎟⎟ cos nθ n 2n 2 h ⎝⎝ ⎠ ⎝ ⎠ ⎠

ω 2ρ f R

Actually, the r dependency can be removed, based on the approximation: n

-n

x⎞ ⎛ x⎞ ⎛ ⎜1 + ⎟ + ⎜1 + ⎟ 2 ∀n , 0 ≤ x ≤ h ⎝ R⎠ ⎝ R⎠

whence the simplified form: pn (θ ) =

ω 2ρ f R n 2h

( u ( 2 ) − u (1) ) cos nθ n

[2.160]

n

The added mass coefficients [2.156] are approximated as: (a) n

m

(1,1) m

(a) n

ρ f π R2 R ( 2, 2 ) 2 n h

;

(a) n

m

(1, 2 ) = m

(a) n

ρ f π R2 R [2.161] ( 2,1) − 2 n h

It is worth noticing that the approximate added mass matrix is null, as easily checked by calculating the eigenvalues, which are found to be λ1 = 0; λ2 = 2 . If the shells have the same mechanical properties, the in-phase coupled mode is such that un ( 2 ) = un (1) and the modal added mass is zero. At the opposite, the out-of-phase mode is such that un ( 2 ) = −un (1) and the modal added mass takes on the large value: mn( a ) =

2ρ f π R2 R n2 h

[2.162]

Inertial coupling

99

The simplifications made just above can be introduced in a distinct and interesting manner, as follows. Starting from the equations [2.133], transformed here into: ∂ 2 pn 1 ∂ pn n 2 + − pn = 0 ∂r 2 r ∂ r r 2 ∂ pn ∂ pn = ω 2 ρ f un (1) cos nθ ; r R = 1 ∂r ∂r

[2.163] 2

r = R2

= ω ρ f un (2) cos nθ

The idea is to replace the local pressure pn ( r,θ ) by its mean value pn (θ ) , as averaged through the fluid layer thickness: pn (θ ) =

R+h

1 ⌠⎮ h ⎮⌡R

pn ( r,θ ) d r

[2.164]

For that purpose, we start by integrating the Laplacian in the radial direction: R+h

⌠ ⎮ ⎮ ⎮ ⎮ ⌡R

⎛ ∂ 2 pn 1 ∂ pn n 2 ⎞ pn ⎟ dr = 0 − ⎜ 2 + r ∂ r r2 ⎠ ⎝ ∂r

[2.165]

The first term is of particular interest as it gives: R+h

⌠ ⎮ ⎮ ⎮ ⎮ ⌡R

∂ pn ∂ 2 pn dr = 2 ∂r ∂r

r =R+h

−

∂ pn ∂r

r=R

= ω 2 ρ f ( un (2) − un (1) ) cos nθ

[2.166]

The physical meaning of the result [2.166] is clear, as it stands for the source term induced in the fluid equation by the motion of the shells. The other terms are integrated as follows: R+h

⌠ ⎮ ⎮ ⎮ ⎮ ⌡R

⌠

R+h

⎮ ⎛ 1 ∂ pn n 2 ⎞ − 2 pn ⎟ dr = ⎮⎮ ⎜ ⎝r ∂r r ⎠ ⎮

⎛ ∂ ⎛ pn ⎞ 1 − n 2 ⎞ ⎜ ⎜ ⎟ + 2 pn ⎟ dr r ⎝∂r ⎝ r ⎠ ⎠

⌡R

R+h

⌠ ⎮ ⎮ ⎮ ⎮ ⌡R

R+h

⌠ ⎮ ⎮ ⎮ ⎮ ⌡R

∂ ⎛ pn ⎞ h ⎜ ⎟ dr = − 2 pn ∂r⎝ r ⎠ R

[2.167] ⌠

⎮ 1 − n2 pn dr (1 − n 2 ) pn ⎮ 2 ⎮ r ⎮

R +h

⌡R

1 h dr = (1 − n 2 ) 2 pn 2 r R

100


Collecting the partial results [2.166] and [2.167], we arrive at a mean pressure field which is identical to [2.160] and then to the same added mass coefficients as those given by [2.161]. 2.3.3.2 Extension to other geometries The approximation introduced in the last subsection is in agreement with the dimensional analysis presented in Chapter 1, subsection 1.3.3.1 and can be extended to any geometry giving rise to a thin layer of fluid. Let us consider for instance the case of a rectangular plate which vibrates with the amplitude Z s ( x, y ) near a rigid wall in the presence of an interstitial layer of liquid. The layer thickness h ( x, y ) is assumed to be small in comparison with the length a and the width b of the plate and the fluctuating pressure is assumed to vanish at the edges of the plate. The local pressure induced by the vibration is governed by the 3D equation: ∂2 p ∂2 p ∂2 p + + = −ω 2 ρ f Z s ( x, y ) δ ( z − h ( x, y ) ) ∂x 2 ∂y 2 ∂z 2

[2.168]

p ( 0, y ) = p ( a, y ) = p ( x,0 ) = p ( x, b ) = 0

The mean pressure field is defined as: p ( x, y ) =

1 ⌠h ⎮ p ( x , y , z )dz h ( x, y ) ⎮⌡0

[2.169]

By integrating [2.169] over the fluid layer thickness, we arrive at the 2D equation: ω 2 ρ f Z s ( x, y ) ∂2 p ∂2 p + = − ∂x 2 ∂y 2 h ( x, y )

[2.170]

The general solution of the problem [2.170] in the absence of vibration ( Z S = 0 ) is easily found by using the separation of variables method as: n =+∞

m =+∞

∑ ∑α n =1

m =1

n ,m

⎛ nπ x ⎞ ⎛ mπ y ⎞ n =+∞ sin ⎜ ⎟ sin ⎜ ⎟= ∑ ⎝ a ⎠ ⎝ b ⎠ n =1

m =+∞

∑α m =1

n ,m

pn , m ( x , y )

[2.171]

where α n ,m are arbitrary constants and pn ,m ( x, y ) are the eigenfunctions of the Laplacian provided with pressure nodes at the edges of the fluid layer. Solution of the problem [2.170] in the presence of vibration is more or less simple depending on the boundary conditions to be fulfilled at the plate edges. Assuming, as a first example, a plate hinged at the four edges and a fluid layer of uniform thickness h, the modes shapes of the plate are the same as the pressure eigenfunctions, see [AXI 05], Chapter 6. In that case, solution is immediate, due to the orthogonality of the mode shapes. By substituting the pressure [2.171] into the

Inertial coupling

101

equation [2.170] and projecting the result onto the mode shape, it is readily found that: α n ,m =

(

ω 2 ρ f ( ab )

2

[2.172]

)

π 2 ( nb ) + ( ma ) h 2

2

The generalized force exerted by the fluid on the plate follows as: a

Q ( n, m )

⌠ ⎮ = α n ,m ⎮⎮ ⎮ ⎮ ⌡0

⌠

⎛ ⎛ nπ x ⎞2 ⎞ ⎮⎮ ⎜⎜ sin ⎜ ⎟ ⎟⎟ dx ⎮ ⎝ ⎝ a ⎠ ⎠ ⎮⎮

b

⌡0

⎛ ⎛ mπ y ⎞2 ⎞ ab ⎜⎜ sin ⎜ ⎟ ⎟⎟ dy = α n ,m 4 ⎝ ⎝ b ⎠ ⎠

The added mass is thus: M a ( n, m ) =

(

ρ f ( ab )

3

[2.173]

)

4π 2 ( nb ) + ( ma ) h 2

2

In this case, the mode shapes are not modified by fluid coupling. As a second example, we consider a rigid plate vibrating according to a pure translation mode: Z s = Z 0 , where Z 0 is the arbitrary magnitude of the motion. Here a difficulty arises as the mode shape differs from the pressure eigenfunctions. However, the problem can still be solved analytically, by using either an expansion of Z s as a double Fourier series, or by using the Rayleigh-Ritz (or Galerkin) procedure, already described in [AXI 05], Chapter 5. It turns out that, in the present example, the two methods are equivalent to each other because the Fourier series is compatible with the pressure eigenfunctions. Using the Rayleigh-Ritz procedure, the substitution of the pressure field [2.171] into the equation [2.170] gives: n =+∞

∑ n =1

⎛ ( nb )2 + ( ma )2 ⎞ ⎛ nπ x ⎞ ⎛ mπ y ⎞ ω 2 ρ f Z 0 sin = α n ,mπ ⎜ ⎟ sin ⎜ ∑ 2 ⎜ ⎟ ⎝ a ⎟⎠ ⎜⎝ b ⎟⎠ h m =1 ( ab ) ⎝ ⎠

m =+∞

2

[2.174]

Interpreting the eigenfunctions as admissible trial functions, the equation [2.174] is projected onto pn ,m to obtain the coefficients α n ,m as the solution of: ⎛ ( nb )2 + ( ma )2 ⎞ π 2 ab ω 2 ρ f Z 0 α n ,m ⎜ = ⎟ 2 ⎜ ⎟ 4 h ( ab ) ⎝ ⎠

It is readily found that:

a

⌠ ⎮ ⎮ ⎮ ⎮ ⌡0

⌠

b

⎛ nπ x ⎞ ⎮⎮ ⎛ mπ y ⎞ sin ⎜ ⎟ dx ⎮ sin ⎜ ⎟ dy ⎝ a ⎠ ⎮ ⎝ b ⎠ ⌡0

[2.175]

102


α n ,m =

2 n m 4ω 2 ρ f Z 0 ( ab ) ⎛ (1 − ( −1) )(1 − ( −1) ) ⎞ ⎜ ⎟ 2 2 ⎜ ⎟ hπ 4 nm nb ma + ( ) ( ) ⎝ ⎠

[2.176]

Figure 2.30. Pressure field induced by the plate vibration, with a = 2 m, b = 1 m, fluid layer thickness 5 cm

Projection of the pressure field on the plate displacement gives the added mass coefficient: Ma =

4 ρ f ( ab ) hπ

6

3

n =+∞

m =+∞

n =1

m =1

∑ ∑

⎛ (1 − ( −1) n )(1 − ( −1) m ) ⎞ ⎜ ⎟ ⎜ ⎟ nm ⎝ ⎠

2

⎛ ⎞ 1 ⎜ ⎟ 2 2 ⎜ ( nb ) + ( ma ) ⎟ ⎝ ⎠

[2.177]

Figure 2.30 shows the pressure field induced by the plate motion, as computed by summing the series up to nmax = mmax = 10 , which is sufficient to obtain an

accuracy better than 1% in the result. Figure 2.31 shows the ratio μ a ( b / a ) of the actual added mass to the value given by the strip model: M as = ρ f

ba 3 12h

[2.178]

Proof of the formula [2.178] is left to the reader as a short exercise. As the aspect ratio b/a decreases, μ a diminishes, almost linearly in the range 0.3 K c2

; M 1 M 2 > M c2

[2.191]

The equation giving the natural frequencies of [2.189] is written in terms of λ = ω 2 , as:

(M M 1

2

− M c2 ) λ 2 − ( K 2 M 1 + K1 M 2 − 2 K c M c ) λ + K1 K 2 − K c2 = 0

[2.192]

Using [2.191] and the fact that M c is negative, it can be concluded that the roots of [2.192] cannot be negative. Furthermore, they are necessarily distinct from each other, as proved by looking at the discriminant: D = ( K 2 M 1 − K1 M 2 ) + 4 ( M c2 K1 K 2 + K c2 M 1 M 2 − K c M c ( K 2 M 1 + K1 M 2 ) ) 2

[2.193]

Using again [2.191], the second term of [2.193] can be conveniently simplified to show that D is always positive: M c2 K1 K 2 + K c2 M 1 M 2 − K c M c ( K 2 M 1 + K1 M 2 ) > 2 M c2 K c2 − 2 M c2 K c2 = 0

[2.194]

Figure 2.41. Frequency plots of the shell modes n = 7, m = 1: simplified model

114


It is worth emphasizing that a negative value of D would mean that the roots of [2.192] are complex conjugate. Hence, they would not describe harmonic motions, but oscillations whose magnitude vary as an exponential function of time, as already discussed in [AXI 04], Chapter 5. Figure 2.41 illustrates the results of the simplified model [2.189], which is in qualitative agreement with the essential features of the experimental data and FEM model. Of course, to obtain a satisfactory fit to the experimental data it would be necessary to account for the modifications of the coupled mode shapes with the water height and for the presence of a pressure node at the free surface. These aspects of the problem are considered in the next subsection, based on an example more amenable to analytical calculation than the present one. 2.3.4.3 Water tank with flexible lateral walls

Figure 2.42. Water tank filled with water: strip model

The relative importance of a pressure node at the free surface of the liquid and the coupling of the structural modes in vacuum due to the fluid pressure are illustrated here in the system sketched in Figure 2.42, which can be solved by a semi-analytical calculation based on the Rayleigh-Ritz method. The structure is a water tank of rectangular cross-section partially filled with water. The length of the tank in the direction perpendicular to the plane of the figure is supposed sufficiently large to allow us to use the strip model. The bottom of the tank is fixed and the lateral walls are assumed to be hinged at their edges z = 0 and z = L. Thus, in the present exercise, we use the following mode shapes of the lateral plates in vacuum:

Inertial coupling

⎛ nπ z ⎞ Φ n ( z ) = sin ⎜ ⎟ ⎝ L ⎠

115

[2.195]

These modes are coupled by the fluid inertia, giving rise to in-phase and out-ofphase coupled modes. Due to the symmetry of the system with respect to the midplane x = 0, the fluctuating pressure related to the in-phase modes vanishes necessarily at x = 0, while the lateral component (Ox) of the pressure gradient related to the out-of-phase modes must vanish at x = 0. Therefore, the problem can be conveniently split into two simpler and similar problems, defined in half of the real system only, as sketched in Figure 2.42. Here, we treat the case of the in-phase modes only. The case of the out-of-phase modes would be solved in a quite similar way. The fluid is governed by the following boundary value problem: ∂2 p ∂2 p + =0 ∂x 2 ∂z 2 n =+∞ ∂p ⎛ nπ z ⎞ = ω 2 ρ f X s ( z ) = ω 2 ρ f ∑ qn sin ⎜ ⎟ ∂x x = a ⎝ L ⎠ n =1 p ( x, H ; ω ) = 0;

p ( 0, z; ω ) = 0;

∂p ∂z

[2.196]

=0 z =0

The pressure field which comply with the homogeneous boundary conditions can be written as: p ( x , z; ω ) =

k =+∞

∑b k =0

k

⎛ ( 2k + 1) π z ⎞ ⎛ ( 2k + 1) π x ⎞ cos ⎜ ⎟ sinh ⎜ ⎟ 2H 2H ⎝ ⎠ ⎝ ⎠

[2.197]

The condition at the fluid-structure interface reads as: ω 2ρ f

n =+∞

∑q n =1

n

⎛ ( 2k + 1) π z ⎞ ⎛ ( 2k + 1) π a ⎞ ⎛ nπ z ⎞ k =+∞ ( 2k + 1) π sin ⎜ cos ⎜ ⎟ cosh ⎜ ⎟ [2.198] ⎟ = ∑ bk L 2 H 2 H 2H ⎝ ⎠ k =0 ⎝ ⎠ ⎝ ⎠

Obviously, a difficulty arises in [2.198] as the pressure profile is distinct from the structural mode shape, which invalidates any attempt to solve the problem by separating the variables. However, an approximate solution can be built by applying the Rayleigh-Ritz method to the interface condition. The cosine functions form an orthogonal set of admissible trial functions on the interval [0,H]. Accordingly, the local condition [2.198] is replaced by an integral condition, from which the coefficients bk can be expressed in terms of the generalized displacements qn :

116

Fluid-structure interaction ⌠

H

2

⎛ ( 2k + 1) π a ⎞ ⎮⎮ ⎛ ⎛ ( 2k + 1) π z ⎞ ⎞ ( 2k + 1) π bk cosh ⎜ ⎟ ⎮ ⎜⎜ cos ⎜ ⎟ ⎟⎟ dz = 2H 2H 2H ⎝ ⎠ ⎮⎮ ⎝ ⎝ ⎠⎠ ⌡0

H

⌠ n =+∞ ⎮ 2 ω ρf qn ⎮⎮ n =1 ⎮ ⎮ ⌡0

∑

[2.199]

⎛ ( 2k + 1) π z ⎞ ⎛ nπ z ⎞ cos ⎜ ⎟ sin ⎜ ⎟ dz 2H ⎝ ⎠ ⎝ L ⎠

The result is written as: bk =

8ω 2 ρ f LH Σ n ( k )

⎛ ( 2k + 1) π a ⎞ π 2 ( 2k + 1) cosh ⎜ ⎟ 2H ⎝ ⎠

⎛ k ⎛ nπ H ⎞ ⎜ ( 2k + 1) L ( −1) sin ⎜ L ⎟ − 2nH ⎝ ⎠ Σ n (k ) = ∑ ⎜ 2 2 n =1 ⎜ 2 k 1 L 2 nH + − ) ) ( ) (( ⎜ ⎝ n =+∞

⎞ ⎟ ⎟ qn ⎟ ⎟ ⎠

[2.200]

It is noticed that if ( 2k + 1) L = 2nH , the coefficient of qn simplifies into ( 4nH ) ; −1

whence the wall pressure: 2

p ( a , z; ω ) =

8ω ρ f LH π2

k =+∞

∑ k =0

⎛ ( 2k + 1) π a ⎞ tanh ⎜ ⎟ Σ n (k ) 2H ⎛ ( 2k + 1) π z ⎞ ⎝ ⎠ cos ⎜ ⎟ 2H ( 2k + 1) ⎝ ⎠

[2.201]

Once more, the added mass coefficients are obtained by projecting the wall pressure on the structural mode shapes: H

Qm =

⌠ ⎮ ⎮ ⎮ ⎮ ⌡0

n =+∞ ⎛ mπ z ⎞ 2 p ( a, z; ω ) sin ⎜ ⎟ dz = ω ∑ M a ( m, n ) qn ⎝ L ⎠ n =1

where the added mass coefficient is expressed as:

[2.202]

Inertial coupling

M a ( m, n ) =

117

8 ρ f LH

S ( n, m ) π2 ⎛ ⎛ ( 2k + 1) π a ⎞ ⎛ k ⎞⎞ ⎛ nπ H ⎞ ⎜ I ( k , m ) tanh ⎜ ⎟ ⎜ ( 2k + 1) L ( −1) sin ⎜ ⎟ − 2nH ⎟ ⎟ k =+∞ 2 H L ⎝ ⎠ ⎠⎟ ⎝ ⎠⎝ S ( n, m ) = ∑ ⎜ 2 2 ⎜ ⎟ k =0 ( 2k + 1) ( ( 2k + 1) L ) − ( 2nH ) ⎜⎜ ⎟⎟ ⎝ ⎠ H mπ H ⎞ k ⎛ ⌠ mH − ( k + 1/ 2 ) L ( −1) sin ⎜ ⎟ ⎮ + 2 1 k π z ⎛ ⎞ ( ) m π z LH ⎛ ⎞ ⎝ L ⎠ I ( k , m ) = ⎮⎮ cos ⎜ ⎟ sin ⎜ ⎟ dz = 2 2 2H π ⎮ ( mH ) − ( ( k + 1/ 2 ) L ) ⎝ ⎠ ⎝ L ⎠ ⎮

)

(

⌡0

[2.203] As a first application, we consider the case of steel walls of thickness e = 1 cm v and length L = 4 m. The natural frequencies f n( ) of the bending modes in vacuum are reported in Table 1. Table 1. Natural frequencies of the in-phase modes

n:

1

(v)

f n (Hz) :

1.417

2

3

4

5

6

7

5.668 12.75 22.67 35.43 51.12 69.43

f n( ) (Hz) : 0.5062 2.965 6.280 11.86 20.05 28.67 42.23 w

M n( ) / M n( w

v)

8.215 2.612 1.684 2.907 1.656 2.210 1.847

To calculate the modes when the tank is filled with water, the semi-analytical model described just above was implemented in MATLAB. The modal basis is truncated up to n = 15 and the set of the trial functions, used to describe the fluid, is w truncated up to k = 50. The computed natural frequencies f n( ) of the first seven inphase modes of the tank filled up to mid-height H = 2 m, are also reported in Table 1, together with the ratios of the modal masses. On the other hand, the higher the modal index, smaller is the vibration amplitude of the wetted part of the wall, as illustrated in Figure 2.43, which displays the mode shapes n = 1 and n = 5 of a wall together with the fluid motion. Such a result can be qualitatively understood since the inertia forces induced by the fluid increase as the square of the modal frequency, hence as n 4 . However this trend is modified to some extent by the n-dependency of the modal mass, see third row of Table 1. Turning now to the fluid oscillation, in the same way as in Figures 2.25 and 2.26, the full lines stand for the isobars and the arrows for the fluid acceleration. As expected, the magnitude of the pressure fluctuations steadily decreases along the Ox direction, from the wall to the midplane, whereas along the Oz direction a wavy profile is observed with pressure crests and troughs in accordance with the wall oscillation.

118


Figure 2.43. Mode shapes of the wall and associated fluid oscillation

Furthermore, the pressure profile is also affected by the pressure node at the free surface and the pressure gradient node at the bottom of the tank; whence the asymmetrical shapes of the isobars. Of course, the fluid flows from the high pressure to the depressed regions. In particular, referring to the n = 1 case, there is a significant inward flow from the free surface to satisfy both the condition of fluid

Inertial coupling

119

incompressibility and volume change associated with the wall vibration. Fluctuating pressure and flow are changed in sign in the complementary part of the tank −a ≤ x ≤ 0 . To conclude on this part of the exercise, it is worth emphasizing that the results of the semi-analytical model were found to be in very close agreement with those obtained by using a finite element model. Finally, the present example can be used also to investigate the effect of the pressure node at the free surface. To alleviate the mathematical analysis, it is found convenient to restrict the discussion to the particular case H= L. It turns out that in the range of large values of the aspect ratio L / a >> 1 , the general result [2.203] can be drastically simplified as follows: ⎛ ( 2k + 1) π a ⎞ ( 2k + 1) π a tanh ⎜ ⇒ ⎟ 2L 2L ⎝ ⎠ ⎛ 32 ρ f aL k =+∞ ⎜ nm M a ( m, n ) = ∑ 2 2 2 2 2 ⎜ π k = 0 ⎜ ( 2n ) − ( 2k + 1) ( 2m ) − ( 2k + 1) ⎝

(

)(

)

⎞ ⎟ ⎟⎟ ⎠

[2.204]

Furthermore, it can be verified that the series vanishes if n ≠ m and so the final result reads as: ⎧ ρ f aL if m = n ⎪ M a ( n, m ) = ⎨ 2 ⎪⎩ 0 otherwise

[2.205]

Figure 2.44. Ratio of the “real” added mass coefficient to the simplified value [2.205]

120


The formula [2.205] is quite remarkable for its simplicity and can be immediately identified with that which would arise from the strip model along the Oy direction. Indeed, as the walls are assumed to vibrate in-phase, the added mass of a fluid strip extending from z to z+dz and from x = 0 to x = a, is equal to the displaced mass dma = ρ f adz . Thus, the added mass per unit length of tank (strip model along the length direction) vibrating according to a mode shape [2.195] is equal to the value given by the formula [2.205]. Now, it can be understood that the departure between the asymptotic value and the “real value” which accounts for the pressure node on the free surface, is entirely contained in the hyperbolic tangent term. Figure 2.44 illustrates the big importance of the pressure node at the free surface to reduce the added mass coefficient with respect to the value obtained by adopting the strip model. The effect is found to increase substantially with the mode index and the reciprocal of the aspect, or slenderness, ratio. 2.3.5

3D problems

2.3.5.1 Plate immersed in a liquid layer of finite depth

Figure 2.45. Rectangular plate immersed in a liquid layer

As sketched in Figure 2.45, we consider an horizontal liquid layer of uniform thickness H, bounded by a free surface at z = H1 and by a rigid bottom at z = − H 2 . At z = 0 a thin rectangular plate of length a and width b is assumed to vibrate vertically, according to the transverse mode shapes: ⎛ nπ x ⎞ ⎛ mπ y ⎞ Φ n ,m = sin ⎜ ⎟ sin ⎜ ⎟ ⎝ a ⎠ ⎝ b ⎠

[2.206]

Inertial coupling

121

The fluctuating pressure is supposed to vanish outside the domain [0 ≤ x ≤ a; 0 ≤ y ≤ b] . The fluid problem is split into two distinct domains according to the face of the plate which is concerned. It is thus formulated as: Domain 0 ≤ z ≤ H1 ; 0 ≤ x ≤ a ; 0 ≤ y ≤ b ⎧ ∂ 2 p1 ∂ 2 p1 ∂ 2 p1 + 2 + 2 =0 ⎪ ∂x 2 ∂y ∂z ⎪⎪ p 0, y , z = p a , y , z = p x , 0, z ) = p1 ( x, b, z ) = 0 ( ) ( ) ( ⎨ 1 1 1 ⎪ ∂p1 ⎪ p1 ( x, y, H1 ) = 0; = ω 2 ρ f Z s ( x, y ) ∂z z = 0 ⎪⎩

[2.207a]

Domain − H 2 ≤ z ≤ 0 ; 0 ≤ x ≤ a ; 0 ≤ y ≤ b ⎧ ∂ 2 p2 ∂ 2 p2 ∂ 2 p2 + + 2 =0 ⎪ ∂x 2 ∂y 2 ∂z ⎪ ⎪ ⎨ p2 ( 0, y , z ) = p2 ( a, y, z ) = p2 ( x,0, z ) = p2 ( x, b, z ) = 0 ⎪ ∂p2 ∂p2 ⎪ =0 ; = −ω 2 ρ f Z s ( x, y ) ∂z z =− H 2 ∂z z = 0 ⎪⎩

[2.207b]

Solving the problem [2.207] is a straightforward task by separating the variables. No coupling occurs between distinct modes of vibration and the pressure field associated with the mode (m,n) is found to be: p1 ( x, y, z; ω ) = ω 2 ρ f

sinh ( kn ,m ( H1 − z ) ) kn ,m cosh ( kn ,m H1 )

p2 ( x, y , z; ω ) = −ω 2 ρ f 2

⎛ nπ x ⎞ ⎛ mπ y ⎞ sin ⎜ ⎟ sin ⎜ ⎟ ⎝ a ⎠ ⎝ b ⎠

cosh ( kn ,m ( H 2 + z ) ) kn ,m sinh ( kn ,m H 2 )

⎛ nπ x ⎞ ⎛ mπ y ⎞ sin ⎜ ⎟ sin ⎜ ⎟ ⎝ a ⎠ ⎝ b ⎠

[2.208]

2

⎛ nπ ⎞ ⎛ mπ ⎞ where kn ,m = ⎜ ⎟ +⎜ ⎟ . ⎝ a ⎠ ⎝ b ⎠

The added mass coefficient follows as: ⎧ ⎛Hπ ⎪ 1 ρ f ( ab) ⎨tanh ⎜ ⎜ ⎪⎩ ⎝ M a ( n, m) = 2

( bn ) + ( am) 2

ab 4π

2

⎞ ⎛H π ⎟ + coth ⎜ 2 ⎟ ⎜ ⎠ ⎝

( bn ) + ( am) 2

2

( bn ) + ( am) 2

ab

2

⎞⎫ ⎟ ⎬⎪ ⎟⎪ ⎠⎭

[2.209]

The result [2.209] brings out the importance of the boundary conditions at the free surface and the fixed bottom. Moreover, it is well suited to help discuss briefly

122


a few particular cases of practical interest. First, the case of an infinite extent of water can be recovered by letting H1 and H 2 tend to infinity. The corresponding added mass coefficient is thus: M a ( n, m ) =

ρ f ( ab )

( bn )

2π

2

+ ( am )

2

[2.210]

2

In practice, the influence of the boundary conditions vanishes if H1 , or H 2 , is

larger than a few characteristic lengths λn ,m ( a, b ) , closely related to the longitudinal and lateral wavelengths of the plate vibration: λn ,m =

ab π

( bn )

2

+ ( am )

[2.211]

2

Then, if H = H1 + H 2 tends to zero, the inertia of the upper fluid layer vanishes, while that of the lower layer becomes very large, due to the confinement effect. In agreement with the formula [2.173], the added mass coefficient becomes: M a ( n, m ) =

ρ f ( ab )

4π

2

(( bn )

2

2

+ ( am )

2

)

⎛ ab ⎞ ⎜ ⎟ ⎝ H2 ⎠

[2.212]

Finally, the case of H1 small and H 2 very large is also of interest to stress again the effect of immersion depth of the plate: M a ( n, m ) =

ρ f ( ab ) ⎧⎪ ⎨ H1 + 4 π ⎪⎩

ab

( bn )

2

+ ( am )

2

⎫ ⎪ ⎬ ⎪⎭

[2.213]

2.3.5.2 Circular cylindrical shell of low aspect ratio We come back to the problem treated in subsection 2.3.2.1 by using the strip model. As clearly indicated by the results presented in subsections 2.3.4.3 and 2.3.5.1, the validity of such a model is questionable if the height H of the cylinder is not very large in comparison with the radius R. Hence, it is of interest to address this point by accounting for the axial variation of the fluctuating pressure field. The problem is solved here for a cylindrical tank filled with liquid up to the top, where it is bounded by a free surface. The bottom of the tank is fixed. The shell vibrates according to a radial mode shape of the type U ( z,θ ) = U n ( z ) cos nθ . The fluid oscillations are governed by the following boundary value problem:

Inertial coupling

∂ 2p 1∂ p 1 ∂ 2p ∂ 2p + + + =0 ∂ r2 r ∂ r r2 ∂ θ 2 ∂ z2 ∂p = ω 2 ρ f U ( z ) cos nθ ∂ r r=R

∂p ∂z

;

123

[2.214] =0

;

p(H) = 0

z =0

We start by attempting to solve the problem by separating the variables. Thus the pressure is written as:

a

f af af af

p r, θ , z = A r B θ C z

[2.215]

leading to: A ′′ A ′ B ′′ C ′′ + + + = 0 ⇒ C ′′ + k 2 C = 0 A rA r 2 B C

[2.216]

Negative values of the constant k 2 are appropriate for C(z) to comply with the axial boundary conditions. The following solutions are found: C j ( z ) = cos (α j z ) where α j = k j =

(2 j − 1)π 2H

j ≥1

[2.217]

af

On the other hand, B θ is necessarily of the type Bn (θ ) = cos nθ , to satisfy the condition at the fluid-structure interface. So, equation [2.216] reduces to the ordinary differential equation: A′′ +

A′ ⎛ 2 n 2 ⎞ − ⎜α j + 2 ⎟ A = 0 r ⎝ r ⎠

[2.218]

It turns out that the general solution of [2.218] can be expressed in terms of the modified Bessel functions of the first and the second kind: yn (α j r ) = aI n (α j r ) + bK n (α j r ) NOTE:

[2.219]

Bessel functions

The relation of Bessel functions with the cylindrical geometry is as close as that of sine and cosine functions with circular geometry. Consequently, they are often termed cylindrical functions as sine and cosine are often termed circular functions. Essentials of Bessel functions can be found in many textbooks on applied mathematics, the reader may be referred in particular to [ANG 61], [BOW 58] and for a comprehensive treatise to [WAT 95]. The few properties of interest in the present book are summarized in Appendix A4. Modified Bessel functions of the first kind, denoted here I n x have a finite value at x = 0 while modified Bessel

af

124


af

functions of the second kind, denoted here K n x , tend to infinity as x tends to zero. They are related to the Bessel functions through the following formulas: I n ( x ) = ( −i ) J n ( ix ) ; K n ( x ) = n

af af

π n +1 i ( J n ( ix ) + iYn ( ix ) ) 2

[2.220]

J n x , Yn x denote the Bessel functions of index n, of the first and the second kind, respectively. They are defined as the two independent solutions of the Bessel differential equation: y ′′ +

1 y′ + x

F Fα GH GH

2

−

n2 x2

II y = 0 JK JK

[2.221]

which has thus the general solution: y ( x ) = aJ n (α x ) + bYn (α x )

[2.222]

In the present problem, the physically acceptable solution is the function of the first kind of index n: Aj ( r ) = I n (α j r )

[2.223]

Therefore the fluctuating pressure field is written as the series: ∞

pn ( r,θ , z ) = cos nθ ∑ a j I n (α j r ) cos (α j z ) j =1

[2.224]

af

Of course, there is no reason why the axial function U z appearing in the radial mode shape of the shell would fit to the axial shape [2.224] of the pressure field. Fortunately, U z can be expanded as a series in terms of the functions cos α j z :

af

d i

U ( z ) = ∑ β j cos (α j z )

[2.225]

j =1

Since the base functions are orthogonal in the interval [0,H], the condition at the fluid-structure interface can be expressed term by term as: a jα j I n′ (α j R ) = ω 2 ρ f β j

[2.226]

To illustrate the importance of the pressure node at the free surface, it suffices to assume a simple function for U z . Here we adopt a constant: U ( z ) = 1 , so

af

U n ( z,θ ) = cos nθ , which is expanded as:

Inertial coupling

⎛ 4 ∞ ( −1) j ⎞ U n ( z,θ ) = ⎜ ∑ cos (α j z ) ⎟ cos nθ ⎜ π j =1 2 j − 1 ⎟ ⎝ ⎠

125

[2.227]

We must recognize that only the case n = 1 may be realistically excited as it corresponds to the horizontal mode of translation of a rigid tank. However, due to its simplicity, the mathematical analysis can be extended to the cases n ≠ 1 , which can be achieved only by artificial excitation but have still the academic interest of illustrating how the sensitivity of the added mass coefficients to a free surface changes with the modal rank, or in other terms with the modal wavelength. The wall pressure is expressed as the series: pn ( R,θ , z ) = ω 2 ρ f

8H cos nθ π2

∞

∑ j =1

( −1)

j

( 2 j − 1)

2

I n (α j R ) I n′ (α j R )

cos (α j z )

[2.228]

Figure 2.46 shows a sample of axial shapes of the pressure field, normalized to the value of the strip model, for distinct aspect ratios H/R, where n differs from zero. Figure 2.46a refers to the circumferential index n = 1 and Figure 2.46b to n = 6. In the latter, the oscillations perceptible on the curve H/R = 12 are a mere consequence of the truncation of the series, ( jmax = 50 ). As expected, the pressure field arising from the 3D model is less than that given by the 2D strip model. The effect is more pronounced as the aspect ratio H/R and the circumferential index of the shell mode decreases. If H/R, or/and n is sufficiently large, the result of the strip model remains valid in most of the fluid, except, of course, in the close vicinity of the free surface. The behaviour of the modal added mass is a direct consequence of such trends. The contribution of the j-th pressure term to the added mass is found to be: Mj =

32 ρ f H π

3

( 2 j − 1)

3

I n (α j R ) ⌠ 2π I n′ (α j R )

⎮ ⎮ ⌡0

H

R ( cos θ ) dθ 2

⌠ ⎮ ⎮ ⎮ ⌡0

( cos (α z ) ) dz 2

j

[2.229]

Finally, the added mass coefficient follows immediately as: Ma =

16 ρ RH 2 π2 f

∞

1

∑ ( 2 j − 1) j =1

3

I n (α j R ) I n′ (α j R )

[2.230]

To mark the difference between the 3D and the strip models, the result [2.230] is normalized by the added mass arising from the strip model M 2 D = ρ f π R 2 H / n , where again n differs from zero. The following added mass ratio is obtained: 16 Hn ⎛H⎞ M μ ⎜ ⎟ = 3D = 3 π R ⎝ R ⎠ M 2D

∞

1

∑ ( 2 j − 1) j =1

3

I n (α j R ) I n′ (α j R )

[2.231]

126


As expected, the added mass arising from the 3D model is less than that given by the 2D strip model and the effect is more pronounced as the aspect ratio H/R and the circumferential index of the shell mode decrease, see Figure 2.47. As an interesting point to assess the validity range of the strip model, it is noted that the relative discrepancy is less than ten per cent as soon as H/R is larger than about 5 for the mode n = 1 and larger than about 1 for the mode n = 8.

(a) circumferential index n = 1

(b) circumferential index n = 6 Figure 2.46. Axial profile of the fluctuating pressure

Inertial coupling

127

Finally, it may be worth noticing that the qualitative trends brought in evidence on the Cartesian geometry in subsection 2.3.5.1 and here on the cylindrical geometry, are qualitatively the same. They reflect the basic fact that the fluid flows from a high pressure zone to the nearest depressed zone, while the shape of the pressure map is partly controlled by the structural vibration and partly by the fluid boundaries. As an extreme and enlightening case, we may consider the breathing mode of the shell. It is recalled that according to the strip model the added mass is infinite, a result which means that no vibration and no 2D oscillating flow are possible due to fluid incompressibility. According to the 3D model, fluid motion becomes nevertheless possible due to the free surface. The problem can be easily studied analytically considering the ideal problem of a shell simply supported at both ends and filled with a liquid limited by a free surface at both ends. The first mode shape of the shell corresponds to the radial displacement: ⎛πz ⎞ U ( z ) = U1 sin ⎜ ⎟ ⎝H ⎠

[2.232]

With the aid of result [2.228], the wall pressure field is written as: p ( R, z ) = ω 2

ρ f HU1 ⎛ I 0 (π R / H ) ⎞ ⎛ π z ⎞ ⎜ ⎟ sin ⎜ ⎟ π ⎜⎝ I 0′ (π R / H ) ⎟⎠ ⎝ H ⎠

[2.233]

The added mass coefficient follows immediately as: ⎛ I (π R / H ) ⎞ ⎛ I 0 (π R / H ) ⎞ 2 M a (1,0 ) = ρ f H 2 R ⎜⎜ 0 ⎟⎟ = ρ f H R ⎜⎜ ⎟⎟ ⎝ I 0′ (π R / H ) ⎠ ⎝ I 1 (π R / H ) ⎠

[2.234]

The result [2.234] can be conveniently reduced in a dimensionless form by using the fluid mass contained in the cylinder as a suitable scaling factor: μ a (1,0 ) = ρ f

H ⎛ I 0 (π R / H ) ⎞ ⎜ ⎟ π R ⎜⎝ I1 (π R / H ) ⎟⎠

[2.235]

As expected, the reduced added mass coefficient increases rapidly with the aspect ratio H/R, see Figure 2.48. When using the formula [2.235] to assess the frequency shift for a realistic shell, care has to be taken that in reality the breathing mode of cylindrical shells involves a coupling between the radial, the axial and the tangential displacement. The relative importance of these three components also varies with the aspect ratio of the shell. Direct comparison is however possible with experimental data, or FEM results, in the case of low aspect ratios (typically H/R

128


less or equal to two) because in that range vibration is essentially radial. At higher aspect ratios, corrections are required because the tangential displacement becomes preponderant in vacuum and not in liquid and this precisely because inertial coupling is purely radial.

Figure 2.47. Added mass ratios versus the aspect ratio of the shell for distinct circumferential indexes of the shell modes

Figure 2.48. Dimensionless added mass coefficient of the breathing mode versus the aspect ratio of the shell

Inertial coupling

129

2.3.5.3 Vertical oscillation of an immersed spherical object We consider a rigid sphere of mass M s maintained by a spring of stiffness coefficient K s acting along the vertical Oz direction, see Figure 2.49, where definition of the spherical coordinates and notations are shown. We want to calculate the natural frequency of this oscillator when the system is immersed in a liquid. It is recalled that the coefficients of the spherical metrics are determined as follows: ds 2 = dr 2 + r 2 dϕ 2 + ( r sin ϕ ) dθ 2 2

⇒

g r = 1 ; gϕ = r ; gθ = r sin ϕ

[2.236]

Substituting the coefficients into the Laplacian [2.119], the latter is expressed in spherical coordinates as: Δp =

1 ⎧∂ ⎛ 2 ∂p ⎞ ∂ ⎛ ∂p ⎞ ∂ ⎛ 1 ∂p ⎞ ⎫ ⎨ ⎜ r sin ϕ ⎟ + ⎜ sin ϕ ⎟+ ⎜ ⎟⎬ r sin ϕ ⎩ ∂r ⎝ ∂r ⎠ ∂ϕ ⎝ ∂ϕ ⎠ ∂θ ⎝ sin ϕ ∂θ ⎠ ⎭ 2

[2.237]

or in an equivalent manner as: Δp =

∂ 2 p 2 ∂p 1 ∂p 1 ∂ 2 p 1 ∂2 p + + 2 + 2 + 2 2 2 2 ∂r r ∂r r tan ϕ ∂ϕ r ∂ϕ ( r sin ϕ ) ∂θ

[2.238]

The boundary value problem is thus written here as: ∂2 p 2 ∂ p 1 ∂p 1 ∂ 2 p 1 ∂2 p + + + + =0 ∂ r 2 r ∂ r r 2 tan ϕ ∂ϕ r 2 ∂ϕ 2 ( r sin ϕ )2 ∂θ 2 ∂p ∂r

2

= ω ρ f Z 0 cos ϕ ;

[2.239]

p → 0 if r → ∞

r = R0

x = r sinϕ cosθ y = r sinϕ sinθ z = r cosϕ

Figure 2.49. Vertical oscillation of a sphere in a liquid

130


At first sight at least, the problem is three-dimensional in nature. As in the case of cylindrical geometry, the general solution of the Laplacian can be obtained in terms of special functions related to the spherical geometry, see Appendix A5. However, the solution of the present problem can be found directly. The particular form of the condition at the fluid-structure interface encourages us to search for a solution of the type: p ( r, ϕ ) = A( r ) cos ϕ

[2.240]

which is independent of the azimuth angle θ. Substituting [2.240] into the Laplacian, the following ordinary differential equation is obtained: A′′ +

2 2 A′ − 2 A = 0 r r

[2.241]

af

Then, a solution of the type A r = αr m is attempted. Substitution into [2.241], gives the possible values for m: ⎧ m = −2 m=⎨ 1 ⎩m2 = +1

[2.242]

The solution which complies with the wall motion and the asymptotic condition of vanishing pressure at infinity is thus: p ( r, ϕ ) = −ω 2 ρ f

R03 Z 0 cos ϕ 2r 2

[2.243]

The generalized force follows as: Q = −ω 2 ρ f

R03 ⌠ 2π Z 0 ⎮⎮ dθ ⌡0 2

π

⌠ ⎮ ⎮ ⌡0

sin ϕ ( cos ϕ ) dϕ 2

[2.244]

whence the added mass coefficient: Ma =

2 M ρ f π R03 = d 3 2

[2.245]

M d is the displaced mass of the fluid.

The natural frequency of the oscillator is: f1( ) = w

1 2π

Ks Ms + Ma

[2.246]

As a short application of the result [2.245], let us consider a spherical body released in a liquid at zero initial velocity. In the case of a massive sphere M s >> M a the initial acceleration is –g, downward and in the case of a very light

Inertial coupling

131

ball M s Ms

[2.247]

The body is prevented floating by anchoring it to the solid bottom by a long rigid cable of length L >> R0 . In this way, an inverted pendulum is obtained, as shown in Figure 2.50. The inverted pendulum presents two particularities of interest, at least. The first is that the fluid provides the system with potential energy, due to the coupling which exists in any pendulum between the weight and the angular displacement θ . Here, the potential is: ⎛ 4π R03 ρ f E0 = ⎜ M s − ⎜ 3 ⎝

⎞ ⎟⎟ g (1 − cos θ ) ⎠

[2.248]

132


If the magnitude of the vibration is very small, in such a way that Lθ remains much smaller than the sphere radius, the inertial effect of the liquid is suitably described by the formula [2.245]. Thus the equation of the linear pendulum is: ⎛ 4π R03 ρ f ⎞ X ⎛ 2π R03 ρ f ⎞ − Ms ⎟ g 0 + ⎜ + M s ⎟ X0 = 0 ⎜⎜ ⎟ ⎜ ⎟ 3 3 ⎝ ⎠ L ⎝ ⎠

[2.249]

The frequency of such a pendulum can be varied not only by modifying its length, but also by changing the mass of the sphere. If M s is negligibly small in comparison with that of the displaced water, the highest possible natural frequency for a given length is obtained, which is equal to: fp =

1 2π

2g L

[2.250]

In the next chapter, an interesting analogy between this result and the sloshing mode of the water in a U tube will be made. Finally, if M s becomes larger than the displaced mass of liquid, the upper position of static equilibrium of the pendulum becomes unstable (the so-called buckling or divergence instability already studied in [AXI 04] and [AXI 05]) and the system can oscillate about the lower position of static equilibrium, provided the fixed point of the pendulum is suitably located above the ground. In such a case, the equation of the linear pendulum becomes: ⎛ 4π R03 ρ f ⎜⎜ M s − 3 ⎝

⎞ X 0 ⎛ 2π R03 ρ f ⎞ +⎜ + M s ⎟ X0 = 0 ⎟⎟ g ⎜ ⎟ 3 ⎠ L ⎝ ⎠

[2.251]

The second particularity of the immersed pendulum is that even if the magnitude of the oscillation is sufficiently small to allow a linearized version of the stiffness term, the linear displacement X 0 can be larger than the sphere radius. If such is the case, validity of the added mass coefficient [2.245] to account for the inertial effect of the fluid can be questioned. To address this point, it is possible to adopt the Lagrangian or the Newtonian approach. Once more, the latter involves more computation than the former, because the kinetic energy of the fluid is easily obtained as shown below. By definition, it is given by the integral: Eκ =

1 ρf 2

Here, (Vf

⌠ ⎮ ⎮ ⎮ ⌡(Vf

)

(V ) dV 2

[2.252]

f

) stands for the volume of liquid, which actually is infinite, and V

f

for the

Eulerian velocity field. As the fluid is incompressible and the flow is potential in nature, it satisfies the equation:

Inertial coupling

div V f = div ⎡⎣ grad Φ ⎤⎦ = ΔΦ = 0

133

[2.253]

Φ is the velocity potential defined in Chapter 1, relation [1.51]. Let us consider the volume integral: ⌠ ⎮ ⎮ ⌡(Vf

)

Φ ΔΦdV = 0

Integrating by parts, we obtain the following relation, which can be viewed as another form of the Green identity already invoked above (cf. relation [2.116]): ⌠ ⎮ ⎮ ⌡(Vf

⌠

)

Φ ΔΦdV = ⎮⎮

⌡(S f

)

⌠ Φ grad.Φnd S − ⎮⎮

⎮ ⌡(Vf

(S )

is the surface bounding (Vf

⌠ ⎮ ⎮ ⎮ ⌡(Vf

f

)

)

( gradΦ ) dV 2

f

=0

[2.254]

and n is the unit vector normal to (S f ) and

directed outward from (Vf ) . Whence the following relation:

)

⌠

(V ) dV = ⎮⎮⎮ ( gradΦ ) dV = ⎮⎮⌡ 2

f

⌡(Vf

2

)

In the present problem (Vf

⌠

(S f )

Φ gradΦ.nd S

[2.255]

) is bounded by the surface (W ) of the solid body and a

concentric sphere of arbitrarily large radius. Since the fluid remains still at infinity, the kinetic energy is given by: Eκ =

ρf 2

⌠ ⎮ Φ gradΦ. nd Sb ⎮ ⌡(W )

[2.256]

On the other hand, Φ is the solution of the following boundary value problem: ΔΦ = 0 ∂Φ ∂r

r − r0 = a

∂Φ = X 0 cos ϕ ; → 0 if r → ∞ ∂r

[2.257]

where r0 ( t ) denotes the position of the spherical body, which obviously changes

with time and where the pole line of the spherical coordinates is along the Ox axis (unit vector i ). Comparing [2.257] to [2.239], the solution [2.243] is readily adapted to the present problem as: R 3 X cos ϕ Φ ( ( r − r0 ) , ϕ ) = 0 0 2 2 ( r − r0 )

[2.258]

134


Substituting [2.258] into [2.256], and remembering that n is pointing toward the centre of the sphere, it is found that: Eκ =

ρ f π R03 X 02

π

⌠ ⎮ ⎮ ⌡0

2

d ⎛ ∂E FI = − ⎜ κ dt ⎝ ∂X 0

( cos ϕ )

2

sin ϕdϕ =

ρ f π R03 X 02 3

⎞ ⎟ = − M a X0 ⎠

=

1 M a X 02 2

[2.259]

The result [2.259] shows that the fluid inertia does not depend on the magnitude of the oscillations. Similarly to the case of the piping system studied in subsection 2.2.2.6, this indicates that any excess of convective inertia in some part of the liquid is exactly balanced by a default of inertia in some other part, as shown by using the Newtonian approach of the problem. On the other hand, as a special case, we consider a translation of the body at constant velocity. From [2.259] it is immediately concluded that no force is exerted on the body. This result is known as d’Alembert’s paradox, though it is a mere illustration of the inertial principle of Galileo, according to which the forces are the same in any inertial (or Galilean) frame. The paradoxical aspect lies in obvious contradiction to experience due to the unavoidable presence of friction and vorticity in real flows. The Newtonian approach to the problem is also of interest as it gives us a good opportunity to introduce an important transformation rule of frames of reference which allows us to describe some unsteady incompressible flows by using a moving coordinate system and to make clear why in the present problem the inertia force is the same whether the nonlinear or the linear version of the Euler equations are used. Following here the presentation given in [PAN 86], let ( Σ ) denote an inertial frame of reference and ( Σ ′) a translated frame moving with the velocity V ( t ) with respect to ( Σ ) . The transformation rules for the coordinates and Eulerian velocity fields are written as: t = t′ r = r ′ + r0 ( t ) ;

t ⌠ r0 ( t ) = ⎮⎮ V (τ )dτ

U ( r ; t ) = U ′ ( r ′; t ′) + V ( t )

[2.260]

⌡0

where the primed quantities refer to ( Σ ′) . As is readily shown, the law of incompressibility is the same in both frames: div ⎡⎣U ( r ; t ) ⎤⎦ = div ⎡⎣U ′ ( r ′; t ′) + V ( t )⎤⎦ = div ⎡⎣U ′ ( r ′; t ′) ⎤⎦ [2.261]

Inertial coupling

135

It must be emphasized that the space partial derivatives in [2.261] refer to the same coordinate system as the quantity to be transformed. However, due to the transformation law [2.260] they can be identified to each other, so in indicial notation: ∂ / ∂ri = ∂ / ∂ri′

[2.262]

On the other hand, if a primed quantity is derived with respect to time t, the chain rules of calculus must be used. For example, the derivative of the velocity field is: ⎛ ∂U ′ ⎞ ∂U ′ ∂U ′ dt ′ ⎛ ∂U ′ ⎞ ⎛ ∂r ′ ⎞ ∂U ′ = + ⎜ ⎟.⎜ −V (t ) . ⎜ ⎟ [2.263] ⎟= ∂t ∂t ′ dt ⎝ ∂r ′ ⎠ ⎝ ∂t ⎠ ∂t ′ ⎝ ∂r ′ ⎠ The transformation law for the time derivative is thus found to be: ∂ ∂ = − V ( t ) .grad ∂t ∂t ′

[2.264]

Turning now to the momentum equation [1.43], in the incompressible case, it reads as: ∂U [2.265] ρf + ρ f U .grad U + grad P − μ f ΔU = 0 ∂t Substituting [2.260] into [2.265] and with the aid of [2.263] and [2.264], we arrive at the following momentum equation, expressed in terms of the primed quantities, except pressure: dV ∂U ′ [2.266] ρf + ρ f U ′.grad U ′ + ρ f + grad P − μ f ΔU ′ = 0 ′ dt ∂t The only difference when passing from [2.265] to [2.266] is the transport inertia term, which may be included into the pressure term as follows: dV [2.267] P ′ = P + ρ f r ′. dt By using P′ instead of P, the momentum equation is found to be identical in both frames ( Σ ) and ( Σ ′) . In the special case of an incompressible and potential flow, the momentum equation can be integrated with respect to the space variables to produce the unsteady Bernoulli equation, written in both frames as:

136


∂Φ 1 2 P + U + =0 ∂t 2 ρf ∂Φ′ 1 2 P ′ 1 2 + U′ + = V ∂t 2 ρf 2

[2.268]

where U is assumed to vanish at infinity and where the integration constant C(t) can be set to zero without loss of generality, because we can add to the velocity potential any arbitrary function of time without changing the velocity field. It is also noted that to comply with the transformation rule [2.260] of velocities, the velocity potential must transform according to the following law: Φ = Φ′ + r ′.U [2.269]

Of course, both potentials satisfy the condition of incompressibility: ΔΦ = 0 ; ΔΦ′ = 0

[2.270]

Once Φ or Φ′ is known, equations [2.268] can be used to determine the pressure field. Application of these general results to the sphere translation is straightforward. The velocity potential in the inertial frame is given by the formula [2.258]. However, for calculating the pressure and the force exerted on the sphere, it is found more convenient to use the reference frame ( Σ ′) in which the body is at rest. Hence the transport velocity is defined as V ( t ) = − X 0 i and the relative pressure field is given by the Bernoulli equation: P′ = − ρ f

∂Φ′ 1 + ρ f X 02 − U ′2 ∂t 2

(

)

[2.271]

Using [2.258] and [2.269], Φ′ is found to be: ⎛ R3 Φ′ = − X 0 ⎜ r − r0 + 0 2 ⎜ 2 ( r − r0 ) ⎝

⎞ ⎛ R3 ⎞ ⎟ cos ϕ = − X 0 ⎜ r ′ + 0 2 ⎟ cos ϕ ⎟ 2r ′ ⎠ ⎝ ⎠

[2.272]

The velocity field follows as: U r′ =

⎛ ⎛ R ⎞3 ⎞ ∂Φ′ = − X 0 ⎜1 − ⎜ 0 ⎟ ⎟ cos ϕ ⎜ ⎝ r′ ⎠ ⎟ ∂r ′ ⎝ ⎠

⎛ 1 ⎛ R ⎞3 ⎞ ∂Φ′ = + X 0 ⎜1 + ⎜ 0 ⎟ ⎟ sin ϕ U ϕ′ = ⎜ 2 ⎝ r′ ⎠ ⎟ r ′∂ϕ ⎝ ⎠

[2.273]

Inertial coupling

137

U r′ vanishes at the body surface ( r ′ = R0 ) and U ′ tends to − X 0i as r ′ tends to infinity, as suitable. The time derivative of the potential yields the acceleration term: ⎛ 1 ⎛ R ⎞3 ⎞ ∂Φ′ = − X0 ⎜ 1 + ⎜ 0 ⎟ ⎟ r ′ cos ϕ ⎜ 2 ⎝ r′ ⎠ ⎟ ∂t ′ ⎝ ⎠

[2.274]

Substituting [2.273] and [2.274] into [2.271], we arrive at: P ′ ( R0 , ϕ ) =

ρf 2 ⎛ 9 3R ρ X 2⎞ X 0 ⎜ 1 − ( sin ϕ ) ⎟ + 0 f 0 cos ϕ 2 2 ⎝ 4 ⎠

[2.275]

Using [2.267], the pressure exerted on the sphere in the inertial frame is found to be: ρ ⎛ ⎞ 2⎞ ⎛ 9 P ( R0 , ϕ ) = P ′ ( R0 , ϕ ) − R0 ρ f X0 cos ϕ = f ⎜ X 02 ⎜ 1 − ( sin ϕ ) ⎟ + R0 X0 cos ϕ ⎟ [2.276] 2 ⎝ ⎝ 4 ⎠ ⎠

The first term, proportional to the body velocity squared, characterizes the pressure due to the convective part of the fluid inertia. It is symmetrical about the equatorial plane, perpendicular to the direction of motion (ϕ = π / 2 ) . The second term, proportional to the body acceleration, characterizes the pressure due to the fluid acceleration; that is the local term of the substantial derivative of the velocity field. It is skew symmetrical about the equatorial plane. The force exerted on the sphere surface is calculated as: ⌠

FI = − ⎮

⌡(Sb )

PndS

⎧ ⌠π ⎫ [2.277] π ⌠ 2⎞ 2 ⎪ ⎮ ⎛ 9 ⎪ FI = −π R ρ f ⎨ X 02 ⎮ ⎜ 1 − ( sin ϕ ) ⎟ cos ϕ sin ϕ dϕ + X0 ⎮⎮ ( cos ϕ ) sin ϕ dϕ ⎬ ⎮ 4 ⎠ ⌡0 ⎪ ⎮⌡0 ⎝ ⎪ ⎩ ⎭ 3 0

The first integral vanishes due to the symmetry of the pressure about the equatorial plane, while the second integral gives the same inertia force as the formula [2.259].

Chapter 3

Surface waves

As explained in Chapter 1, liquid-gas interfaces can oscillate about a static equilibrium due to the restoring forces induced by gravity and surface tension. The oscillations develop as time and space dependent waves which propagate along the interface. They are termed surface waves because, in contrast with the elastic waves developing in solids and in compressible fluids, the oscillatory motion is restricted essentially to a superficial liquid layer one wavelength thick. In the present chapter we will focus first on the gravity waves and their interaction with vibrating solids including floating and grounded flexible structures. Such subjects are of obvious interest in many fields of application concerning ocean and naval engineering. Unfortunately, they are also marked by severe difficulties in mathematical modelling even if restricted to the linear domain. Presentation given here remains thus introductory in nature. The major aspects of gravity waves propagation – which are highly dispersive when travelling on deep water – are first described and illustrated on a few specific examples. Then, standing gravity waves known as sloshing modes and their coupling with vibration modes of structures will be addressed by working out a few moderately simple examples. Based on the oscillatory Froude number, in the earth gravity field, practical importance of such a coupling is essentially restricted to a low frequency range, typically less than about one Hertz. Surface tension is responsible for the occurrence of capillary waves. Based on the Weber number, their relevance as fluid-structure problems is concerned is negligible, in most engineering applications at least. Hence instead interest will be focused on the dynamics of cavitation bubbles. This highly nonlinear dynamical system involving surface tension is of important practical consequences concerning cavitation noise and cavitation erosion. It is classically modelled based on the Rayleigh-Plesset equation. As shown in this chapter, the numerical simulation of expanding and collapsing bubbles may be tricky, and is appropriate to present a short digression into the use of implicit time-integration schemes in the nonlinear domain.

Surface waves

139

3.1. Introduction In Chapter 1, we have seen that gravity and surface tension add potential energy to the free surface separating a dense liquid from a light gas of negligible density. As a consequence, the fluid behaves as a continuum and its motion must be described by using a continuous set of degrees of freedom. In particular, the fluid oscillations about a state of static and stable equilibrium take on the form of waves, known as surface waves. In the present chapter, as in the preceding, the liquid is assumed to be incompressible and non viscous, therefore its motion is still described by the potential flow theory. Moreover, as long as the study is restricted to the linear domain, the fluid-structure coupled problem is still governed by the system [2.1], rewritten here as: M s ⎡⎢ X s ⎤⎥ + K s ⎡⎣ X s ⎤⎦ = − pnδ ( r − r0 ) ⎣ ⎦ Δp = − ρ X .nδ ( r − r ) [3.1] f

s

⎡ σ p(x, y,z) − f ⎢ ρf ⎣⎢

0

⎛ ∂ 3p ∂ 3p ⎞ ∂p ⎤ =0 ⎜ 2 + ⎟+ g ⎥ 2 ∂z ⎦⎥ z = H ⎝ ∂ x ∂z ∂ y ∂z ⎠

where the fluid problem is formulated in terms of fluctuating pressure only and where the results [1.69] and [1.105] are used to express the boundary conditions to be satisfied at the free surface z = H. Once more, either subscript ( f ) or ( s ) is used to stress whether the quantity refers to the fluid or to the solid. On the other hand, depending whether the ratio of the capillary length α f = σ f / ρ f g to the wavelength, is much smaller or, alternatively, much larger than unity, the restoring forces which drive surface waves are due to the weight of the fluid, or to surface tension. As these forces differ in several aspects, it is also appropriate to make the distinction between gravity and capillary waves, as two asymptotic cases at least. In most applications of mechanical engineering the wavelengths of interest largely exceed the capillary length. Therefore, this presentation will focus on gravity waves, although a few salient features of capillary waves will be also briefly described. Section 3.2 is devoted to gravity waves which travel in an infinite extent of liquid whose geometry is marked by an unbounded free surface, while the depth can be either finite, or infinite. For mathematical convenience, the analysis is based on the geometry of a straight canal. The dispersive nature of the travelling waves is first demonstrated and then a few peculiarities of the wave propagation are pointed out in relation to the dimensionless depth parameter η = H / λ , where H stands for the liquid depth and λ for the wavelength. Various curious natural phenomena are shortly introduced, some very devastating, such as the tsunamis and several beautiful sights to be seen every day when walking at any water’s edge, for instance

140


the free surface of a pond impacted by a stone, the wake of a moving ship on open sea, a solitary wave triggered in a canal, etc. Peripheral as such items may appear in relation to the limited objective of this book, it is rewarding to give them a short and introductory presentation, at least. Indeed, much exciting physics can be learned by studying surface waves, which have applications in many other branches of physics and engineering, for instance in oceanography and offshore structural engineering. Section 3.3 is concerned with surface tension. A brief presentation of the capillary waves is first given, and then the importance of surface tension in the cavitation process is emphasized by analyzing the dynamical behaviour of microgas-bubbles in a liquid which act as nuclei for fluid vaporization. Cavitation is a source of acoustical noise and a real concern in mechanical engineering because of the erosive damage it causes. Section 3.4 describes the standing waves which arise in the presence of reflecting boundaries, as the result of the interference between incident and reflected waves, as in the case of the elastic waves in solid bodies. Therefore, referring to the considerations discussed in [AXI 04], Chapter 6 and [AXI 05], Chapter 1, such standing waves can be viewed as natural modes of vibration of the fluid, broadly termed sloshing modes. As the fluid is modelled as a continuum, there is, in principle, an infinite number of such modes. However, in many applications concerned with piping systems the fluid column model can be used, leading to a finite number of sloshing modes. In section 3.5 the coupling between the sloshing and the structural modes is discussed. The practical importance of such a coupling can be suitably assessed based on the oscillatory Froude number [1.101]. As already emphasized in Chapter 1, gravity is significant with respect to inertia in the range F ≤ 1 and so is the coupling between the sloshing and the structural modes. The coupled modes are marked by mode shapes including both fluid and structural components. In the range F >> 1 , coupling is negligible and either the fluid, or the structural component of the mode shape becomes negligible, indicating that the mode may be interpreted in practice as a pure structural, or as a pure sloshing mode. Then a study of the vibrations of floating solids is presented, which concludes the present chapter. 3.2. Gravity waves 3.2.1

Harmonic waves in a rectilinear canal

Let us consider a canal of practically infinite length and of uniform rectangular cross-section. H designates the water depth in the canal and L the width. As shown in Figure 3.1, the Ox axis is along the canal, and Oz is in the upward vertical direction.

Surface waves

141

Figure 3.1. Gravity waves in a rectilinear canal of uniform cross-section

The walls of the canal are supposed to be fixed. In accordance with [3.1], the fluid problem is written as: ∂ 2p ∂ 2p ∂ 2p + + =0 ∂ x2 ∂ y2 ∂ z2 ⎛ ∂ p⎞ p+g =0 ⎜ ⎟ ∂ z ⎠ z=H ⎝

;

∂p ∂z

= z =0

∂p ∂y

= y =0

∂p ∂y

[3.2] =0 y=L

where L designates the width of the canal. To demonstrate that the system [3.2] governs progressive waves which travel along the Ox direction, we seek harmonic solutions, as already explained in [AXI 05] Chapter 1; that is the complex amplitude of pressure is assumed to be: p ( x, y , z; ω ) = po ( x, y , z ) e

i ω t −k .r

(

)

[3.3]

where ω is the circular frequency, r the vector position and k the wave vector, assumed here to be in the Ox direction. In agreement with the convention adopted throughout in the present book, for conservative systems, ω is assumed to be positive, or null eventually. The problem can be further particularized by replacing [3.3] by the much simpler expression: i (ω t − kx ) ⎪⎧ p+ ( z ) e p ( x,z; ω ) = ⎨ i (ω t + kx ) ⎪⎩ p− ( z ) e

x≥0 x≤0

[3.4]

where any dependency in the Oy direction is discarded and where the x-dependency is restricted to the phase term ± kx . The later stands for the phase shift of the state of

142


the wave at a given distance ± x along the direction of propagation with respect to that at the origin, reckoned at the same time t. It is recalled that the phase shift is related to the propagation delay τ by: τ=

x xk = ω cψ

[3.5]

t +τ is the time at which the wave amplitude at ± x is the same as it was at the former time t at x = 0. The phase velocity cψ of the wave is related to the wave number and circular frequency by: cψ =

ω k

[3.6]

The wave p+ , hereafter called an outgoing wave, or forward wave, travels from the left to the right (x > 0) with phase velocity cψ and the wave p− , hereafter called an incoming wave, or backward wave, travels in the opposite direction at the same velocity. Accordingly, kx stands for a phase lag and τ for a time delay whatever the direction of propagation may be, in agreement with the principle of causality, according to which the response of the medium cannot anticipate the excitation. On the other hand, if cψ depends on frequency, the waves are dispersive, that is to say, the spectral components of a polychromatic wave are not travelling at the same phase velocity and the wave profile is modified during propagation, even if the wave is one-dimensional. The mechanical energy conveyed in a dispersive wave propagates at the so called group velocity, which is defined as: cg =

dω dk

[3.7]

Substituting [3.4] into [3.2], the boundary value problem becomes: d 2 p± ( z ) − k 2 p± ( z ) = 0 dz 2 dp −ω 2 p± ( H ) + g ± =0 ; dz z = H

dp± dz

[3.8] =0 z =0

Solution of the differential equation is immediate, giving: p+ ( z ) = p− ( z ) = p0 ( z ) = aekz + be − kz

[3.9]

To comply with the boundary condition at the bottom, the general solution [3.9] must take on the particular form: p ( z ) = α cosh kz

[3.10]

Surface waves

143

Finally, to satisfy the boundary condition at the free surface, the wave number and pulsation must be related to each other by the following dispersion equation: k=

ω2 coth kH ⇔ ω 2 = gk tanh kH g

[3.11]

For a given value of ω , two opposite roots ± k occur which obviously correspond to a pair of an outgoing and an incoming waves. The phase velocity of these waves is found to be: cψ =

g tanh kH tanh kH = g k ω

[3.12]

Hence, it is found that the gravity waves are dispersive and that phase speed increases with the wavelength. The group velocity is: cg =

cψ ⎛ 2kH ⎞ ⎜1 + ⎟ 2 ⎝ sinh 2kH ⎠

[3.13]

The dimensionless quantity kH characterizes the depth of the liquid layer as scaled by the wavelength. The two asymptotic cases of kH tending to zero, and then to infinity, broadly referred to as the shallow water and deep water cases, will be discussed in subsections 3.2.3 and 3.2.5 respectively. The fluid particles follow elliptical orbits with exponentially decreasing radii, as easily demonstrated by considering for instance the wave p+ : p+ ( x, z; ω ) = p0

cosh kz i (ωt − kx ) cosh kz i (ωt − kx ) e e = ρ f gZ 0 cosh kH cosh kH

[3.14]

where Z 0 is the wave amplitude (height of the crests). Substituting [3.14] into the momentum equations, one obtains the complex amplitude of the velocity field of the fluid particles: kgZ 0 cosh kz i (ωt − kx ) ∂ u+ ∂ p + e + = 0 ⇒ u+ = ∂t ∂x ω cosh kH kgZ 0 sinh kz i (ωt − kx ) ∂ w+ ∂ p+ e + = 0 ⇒ w+ = i ρf ∂t ∂z ω cosh kH ρf

[3.15]

Using the relation of dispersion [3.11], u+ and w+ can be more conveniently written as: u+ = ω Z 0

cosh kz i (ωt −kx ) e sinh kH

; w+ = iω Z 0

sinh kz i (ωt − kx ) e sinh kH

[3.16]

144


p+

cψ

z=H

λ

z = H −λ Figure 3.2. Wave profile and particle motion in the wave

By taking the real part of [3.16], the real velocity field is obtained as: cosh kz cos (ωt − kx ) sinh kH sinh kz Re ( w+ ) = −ω Z 0 sin (ω t − kx ) sinh kH Re ( u+ ) = +ω Z 0

[3.17]

As illustrated in Figure 3.2, the fluid particles move on elliptical orbits of parametric equations: cosh kz sin (ω t − 2πα ) sinh kH sinh kz Z p (αλ , z; t ) = Z 0 cos (ω t − 2πα ) sinh kH

X p (αλ , z; t ) = Z 0

[3.18]

where, once more, Z 0 is the height of the wave crest and the coordinates of the ellipse centre are xc = αλ , zc = z . The elliptical orbits can also be described by using the implicit equation: 2

2

⎛ X p ⎞ ⎛ Zp ⎞ 2 ⎜ ⎟ +⎜ ⎟ = Z0 ⎝ A ⎠ ⎝ B ⎠

[3.19]

where the semi-axes a, b and their ratio e are given by: A=

cosh kz sinh kH

; B=

sinh kz sinh kH

; e = tan kH

[3.20]

Surface waves

145

On the other hand, the elevation profile Z + ( x, z; t ) of the wave, as deduced from [3.14], is: Z + ( x, z; t ) = real(

p+ cosh kz ) = Z0 cos (ωt − kx ) ρf g cosh kH

[3.21]

Using [3.17] and [3.21], it is of interest to calculate the mean mechanical energy carried by the wave per unit canal length. Referring to the formula [1.65], the mean potential energy is given by: λ

ep

λ

=

1 1⌠ ρ f gZ 02 L ⎮⎮ cos2 (ω t − kx ) dz λ ⌡0 2

[3.22]

L is the width of the canal (see Figure 3.1) and the angle brackets indicate an average over x, which is performed on a wavelength λ. After a few elementary manipulations, e p λ takes on the remarkably simple form: ep

λ

=

1 1 ⌠ ωt 1 ⎮ cos2 u du = ρ f gZ 02 L ρ f gZ 02 L 2 4 λ k ⎮⌡(ωt − 2π )

[3.23]

In a similar way, the mean kinetic energy is written as: ⌠

eκ

λ

=

λ

H

ρ f L ⎮⎮ 2λ

⌠ ⎮ ⎮ ⎮ ⎮ ⎮ ⌡ 0 ⎮ ⌡0

(u

2

+ w2 ) dzdx =

⎧⎪ ⌠ λ 2 ⎨ ⎮ cos (ω t − kx ) dx 2sinh 2 ( kH ) λ ⎪⎩ ⎮⌡0 ρ f ω 2 Z 02 L

H

⌠ ⎮ ⎮ ⌡0

⌠

λ

cosh 2 ( kz ) dz + ⎮⎮ sin 2 (ω t − kx ) dx ⌡0

H

⌠ ⎮ ⎮ ⌡0

⎫⎪ sinh 2 ( kz ) dz ⎬ ⎪⎭

[3.24] Space averaging is immediate and using then the dispersion relation [3.11], eκ

λ

is

found to be equal to the mean potential energy [3.23]: eκ

λ

=

ρ f ω 2 Z 02 L

4sinh ( kH ) 2

H

⌠ ⎮ ⌡0

cosh 2kz dz =

ρ f gZ 02 L 4

[3.25]

Such a partition of the mechanical energy into equal potential and kinetic parts could be expected a priori, as it is a property common to any linear and conservative oscillating system. 3.2.2

Group velocity and propagation of wave energy

The concept of group velocity cg has already been introduced in [AXI 05], Chapter 1, in relation to elastic waves in solids. It is interesting to verify in the case

146


of gravity waves travelling in a straight canal, that cg is the velocity at which wave energy propagates. With this object in mind, it is appropriate to introduce first a few mathematical entities to describe the transfer of mechanical energy by wave radiation. Recalling that in the case of a particle, mechanical power, or energy flow, is defined as the scalar product of the particle velocity and the force exerted on it, the wave adaptation of this definition uses the concept of wave intensity which is a local quantity, defined as follows: p ( r; t ) X f ( r; t ) .ndS = I ( r ; t ) .ndS [3.26] p ( r ; t ) is the fluctuating pressure of the wave and X f ( r ; t ) is the fluctuating

velocity field of the fluid particles. dS is the area of an elementary surface passing through r and oriented by the unit normal vector n . I ( r ; t ) is known as the instantaneous intensity of the wave, thus defined as: I ( r ; t ) = p ( r ; t ) X f ( r; t )

[3.27]

According to the relation [3.27], the wave intensity is a local and instantaneous entity defined at each position r and time t. The time average of [3.27] yields the mean intensity: T 1⌠ I ( r ) = ⎮⎮ I ( r ; t ) dt T ⌡0

[3.28]

where the over bar denotes a time averaging. The choice of the averaging time T depends on the type of the time profile of the wave being considered. For transient waves, it can be either the duration of the transient, or a characteristic time related to it. For periodic waves, the natural choice is the period. By integrating the intensity over a surface (S ) , one passes from the local to the global scale of energy transfer through through

(S ) . Considering (S ) is defined as:

⌠ P = ⎮⎮ I .nd S ⌡(S )

the mean intensity, the mean wave power radiated

[3.29]

where n is the unit vector, normal to (S ) , and oriented in the direction of wave

propagation. It can be noted that, in particular, no energy flows in a direction perpendicular to the wave intensity vector, or in an equivalent way, to the particle velocity.

Surface waves

147

In the present application, the object is to calculate the mean power radiated through the plane x = 0 by a harmonic surface wave travelling to the right in a straight canal. The mean radiated power is expressed as: Lω 2π

P =

H

2π / ω

⌠ ⎮ ⌡0

dt

⌠ ⎮ ⎮ ⌡0

p+ ( z; t ) u+ ( z; t ) dz

[3.30]

With the aid of [3.17] and [3.21], [3.30] is expressed as: P =

ρ f gZ o2 Lω 2 π sinh 2kH

2π / ω

⌠ ⎮ ⎮ ⌡0

cos2 (ω t ) dt

H

⌠ ⎮ ⎮ ⌡0

cosh 2 ( kz ) dz

[3.31]

Integration is straightforward, it yields: P =

ρ f gZ o2 Lω sinh 2kH

H

⌠ ⎮ ⎮ ⌡0

cosh 2 ( kz ) dz =

ρ f gZ o2 Lω ⎛ 2kH ⎞ ⎜1 + ⎟ 4k sinh 2kH ⎠ ⎝

[3.32]

Substituting the group velocity [3.13] into [3.32] gives the remarkable result: P =

ρ f gZ o2 Lω sinh 2kH

H

⌠ ⎮ ⎮ ⌡0

⎛1 ⎞ cosh 2 ( kz ) dz = ⎜ ρ f gZ o2 L ⎟ cg ⎝2 ⎠

[3.33]

which means that the mean mechanical energy per unit canal length is carried by the wave at the group velocity, as appropriate. 3.2.3

Shallow water waves ( kH > 1)

When the wavelength is much shorter than the water depth, the dispersion equation [3.11] becomes: k=

ω2 ⇒ g

cψ =

g and cg = cψ / 2 ω

[3.78]

Hence, in deep water the gravity waves are dispersive. In contrast with the shallow water waves, the phase and group velocities of the deep water waves are independent of the actual depth of the fluid layer. This case corresponds typically to the short oceanic waves generated by the wind. In Figure 3.5, the wavelength of deep water waves is plotted versus the frequency. The faster the wind, the longer the wind blows, and the bigger the area over which the wind blows, the bigger the waves. In a fully developed sea, most of the power spectral density is contained in a bandwidth extending from about 0.05 Hz up to about 0.15 Hz corresponding to wavelengths roughly within the range from 100 m to 1 km. Wavelengths shorter than a few 10 meters are often termed swash. They are typically observed in almost closed basins and harbours, or ponds and small lakes.

Figure 3.5. Wavelength versus frequency of deep water waves

Surface waves

159

Starting from the complex amplitude [3.14], it is found that the vertical profile of the fluctuating pressure of deep water waves is practically of exponential shape: p ( z; ω ) = po

2π ( H − z ) cosh kZ po exp− λ cosh kH

[3.79]

where again po = ρ f gZ 0 denotes the magnitude of the fluctuating pressure on the free surface and λ the wavelength. Thus a deep water wave is rightly termed a surface wave with the meaning that it is confined in a superficial layer of characteristic thickness λ / 2π much smaller than the depth of the liquid layer. The fluid particles follow circular orbits with exponentially decreasing radius, as immediately confirmed from equation [3.18], or even more directly from the ellipse parameters [3.20]. For instance in a big wave ten meter high and of period ten second, the velocity of the fluid particles is about 6 m/s, which indicates that the impact on a structure, like a dam or the deck of a ship can be devastating, considering the large amount of kinetic energy and linear momentum involved, as further illustrated in subsection 3.2.7. 3.2.5.1 Space and time profiles of progressive waves In deep water, gravity waves can produce nice and intriguing geometrical pictures like those briefly described in Chapter 1, or those associated with the wake of a moving boat, as briefly outlined in the next subsection. Detailed mathematical analysis of such phenomena is not an easy task and does not enter within the scope of a book of this nature. The reader interested in the subject can be referred to a few well known textbooks such as [LAM 32], [LIG 78], [FAB 01]. Here we restrict ourselves to study the profile of a wave triggered by an impulsive source on the surface of a straight canal of infinite depth. The problem was studied for the first time by Cauchy and Poisson [CAU 16]. Let us consider the forward wave which is triggered by an initial displacement field Z o ( x ) , assumed to be an even function of x. As the problem is linear, the wave can be written as the superposition of monochromatic waves of wave number k, which are of the type: pk ( x,z; t ) = α k e k z e

iψ ( k , x )

[3.80]

where the coefficients α k are still unknown and the phase function is as follows: ψ ( k,x ) = ω t − k x = k ( cψ t − x )

[3.81]

The resulting outgoing wave is thus expressed as: ⌠

+∞

p ( x; t ) = ⎮⎮ α k e ⎮ ⌡0

ik ( cψ t − x )

dk

[3.82]

160


On the other hand as Z o ( x ) = Z o ( − x ) , the Fourier transform of the initial displacement can be written as: ⌠

Z 0 ( k ) = ⎮⎮

+∞

⌡−∞

⌠

Z 0 ( x ) e − ikx dx = 2⎮⎮

+∞

⌡0

Z 0 ( x ) cos kx dx

[3.83]

It is noticed that Z 0 ( k ) is also an even function. So, by inverse Fourier transform: Z0 ( x ) =

1 2π

+∞

⌠ ⎮ ⎮ ⌡−∞

+∞

1⌠ Z 0 ( k ) eikx dk = ⎮⎮ Z 0 ( k ) cos kx dk π ⌡0

[3.84]

The expression [3.84] is used to adjust the appropriate coefficient α k in [3.82] by identification. At time t = 0, it follows that: ⌠

+∞

p ( x;0 ) = ⎮⎮ α k e − ikx dk = ρ f gZ 0 ( x ) = ⌡0

ρ f g ⌠ +∞ ⎮ Z ( k ) cos kx dk 0 π ⎮⌡0

[3.85]

whence: αk =

ρ f gZ 0 ( k )

[3.86]

π

Thus the shape of the free surface related to the outgoing wave is given by the integral: Z ( x; t ) =

1 2π

+∞

⌠ ⎮ ⎮ ⎮ ⌡0

ik ( c t − x ) Zo ( k ) e ψ dk

[3.87]

where the factor 1/2 is introduced to account for the symmetry about x = 0 and to the fact that only the outgoing wave is considered. In the case of non dispersive waves (shallow water), the calculation of [3.87] is immediate because the phase term reduces to ψ ( k,x ) = k ( ct − x ) where c = gH . Thus [3.87] reduces to the inverse Fourier transform of Z ( k ) , shifted from the travelled distance ct: 0

⎧1 ⎪ Z ( ct − x ) if x ≤ ct Z ( x; t ) = ⎨ 2 0 ⎪⎩ 0 if x > ct

[3.88]

As may be expected, the result [3.88] fully agrees with those derived in subsection 3.2.4.1. However, in the dispersive case, the calculation is substantially more difficult, even if performed numerically. An approximate solution can be often obtained by using a mathematical technique known as the stationary phase method which closely follows the definition of group velocity. The short presentation made

Surface waves

161

below follows essentially that given in [LAM 32] and [SOM 50]. Let us consider an integral of the kind: ⌠

k2

F = ⎮⎮ f ( k ) e

iψ ( k )

⌡k1

[3.89]

dk

a f varies much less rapidly than the oscillatory function e As a consequence, we can neglect the contribution to the integral of any interval in which f ak f is almost constant. This can be understood intuitively by

where the function f k

a f.

iψ k

referring, either to the chopper property of the oscillatory functions, cf. [AXI 04] Chapter 8, or by referring to the destructive interference process invoked to introduce the concept of group velocity (cf. [AXI 05], Chapter 1). Thus it follows that the major contribution to the integral [3.89] arises from those wave numbers which make the phase function ψ k stationary. Let denote k0 such a value. Expanding ψ k as a Taylor series in the vicinity of k0 , yields:

af

af

ψ ( k ) = ψ ( k0 ) +

( k − k0 )

2

2

ψ ′′ ( k0 ) +

( k − k0 ) 6

3

ψ ′′′ ( k0 ) + O

(( k − k ) ) 4

0

[3.90]

It turns out that an expansion to the second order is sufficient, provided the following condition holds (for mathematical proof, see for instance [LAM 32]): ⎛ ψ ′′′ ( k0 ) ⎞ ⎜⎜ ⎟⎟ ⎝ ψ ′′ ( k0 ) ⎠

3/ 2

1 , the Hankel functions can be conveniently approximated by the asymptotic values, valid for large arguments (see Appendix A4): H n( ) ( kr ) = 1

2 iψ n e π kr

; H n(

2)

( kr ) =

2 −iψ n e π kr

;

⎛ 2n + 1 ⎞ ψ n = kr − ⎜ ⎟π ⎝ 2 ⎠

[7.210] As a consequence, far from the shell, the pressure field [7.209] simplifies into: pn ( r, θ ; t ) =

− i⎛⎜⎜ 2 n+1⎞⎟⎟π / 2 + i⎛⎜⎜ 2 n +1⎞⎟⎟π /2 ⎫⎪ ⎝ ⎠ ⎝ ⎠ cos nθ iωt ⎧⎪ e ⎨( A + iB ) e e −ikr + ( A − iB ) e e + ikr ⎬ kr ⎩⎪ ⎭⎪

[7.211]

where the multipicative constants are tacitly included in the coefficients A and B. In the form [7.211], the pressure wave is suitably split into two distinct travelling waves, as desired. The first component within braces is the outgoing, or diverging wave we are interested in and the second is the incoming or convergent wave coming from infinity. Hence, the anechoic condition at infinity is expressed very simply as: A − iB = 0

[7.212]

Matching of the remaining constant to the fluid-structure interface condition produces the outgoing pressure field excited by the shell vibration, which is expressed as: pn ( r, θ ) = ω ρ f c f

H n(

2)

(2)

H n′

( kr ) U n cos nθ ( kR0 )

[7.213]

Note that the pressure field [7.213] holds whatever the value of kr may be since the exact form of the Bessel or Hankel functions are used. In particular, it can be verified that the generalized force Qn per unit length of the shell exerted by the fluid agrees with the value derived in Chapter 2 provided kR0 is sufficiently small, except for the imaginary part which charaterizes radiation damping. Again Qn is obtained from the functional: ⌠ 2π Qn ,U n cos nθ = − R0U n pn ( R0 ) ⎮⎮ cos2 nθ dθ == π R0U n pn ( R0 ) ⎮ ⌡0

[7.214]

In the range kR0 1) type. As it may be verified on the left-hand plot of Figure 7.43, the radiated power depends on n in the low reduced wave number range, and is independent of n if kR0 is sufficiently large.

Figure 7.43. Power radiated per unit length of a circular cylindrical shell

Of particular interest is the case of a tensioned wire of small diameter, corresponding typically to a musical string instrument. The wire, or string, vibrates transversally according to a circumferential mode n = 1 at a wavelength much larger than the string radius. The radiated power per unit string length is found to be: PR =

ρ f c f π 2 R0 2

( kR0 ) (ω Z 0 ) 3

2

[7.218]

Energy dissipation by the fluid

651

As illustrated in the right-hand side plot of Figure 7.43, a wire of small radius is a very inefficient radiator, which explains why the strings of a musical instrument are always coupled to a resonant cavity. In a violin for instance, coupling is achieved essentially by the bridge which supports the strings and the soundpost located near the bridge which couples belly to the back, both of them acting as soundboards (see for instance [OLS 67], [FLE 98]. 7.2.2

Sound transmission through interfaces

7.2.2.1 Transmission loss at the interface separating two fluids As we have seen earlier, a sound wave impinging on the interface separating two fluids labelled (1) and (2) respectively, is generally partly reflected back and partly transmitted. Therefore, if a sound source lies in fluid (1), part of the sound power is transmitted to fluid (2). If the latter extends to infinity, the sound energy never returns back to fluid (1); which corresponds to a transmission loss. To deal with the problem of determining the transmission loss coefficient, a few important results already established in Chapter 5 subsection 5.1.1.1 are necessary. They are repeated here for convenience using the notations specified in Figure 7.44 which are the same as in Chapter 5. It is recalled that force equilibrium at the interface implies: pi( ) + pr( ) = ptr( ) +

−

+

[7.219]

where pi( ) , pr( ) and ptr( ) stand for the magnitudes of pressure in the incident, reflected and transmitted plane waves, respectively. By using relations [5.6] and [5.10], the condition of continuity of the normal velocities at the interface is written as: +

(

−

cos θi ui( ) + ur( +

−)

+

)

⎛c ⎞ + = uinterface = utr( ) 1 − ⎜ 2 sin θi ⎟ ⎝ c1 ⎠

2

[7.220]

As a definition the pressure reflection coefficient is R = pr( ) / pi( −

(+)

+)

and the pressure

(+)

transmission coefficient is T = ptr / pi . R and T were found to be dependent on the specific impedances of the fluids and on the angle of incidence of the waves, see equation [5.11], repeated here as: R=

Z2(

Z2(

sp ) sp )

cos θi − Z1(

cos θi + Z1(

sp ) sp )

cos θ tr cos θ tr

; T=

Z2(

sp )

2Z2(

sp )

cos θi

cos θi + Z1(

sp )

cos θ tr

[7.221]

652


Figure 7.44. Reflected and transmitted plane waves at the interface between two fluids

It is also useful to characterize reflection and transmission at an interface in terms of radiated power. Using relations [7.94] and [7.95], the power reflection coefficient RP and the power transmission coefficient TP are suitably defined as: RP = TP =

pr2Zi Sr pi2Zr Si ptr2 Zi Str pi2Ztr Si

= =

pr2Z1( ) Si sp

pi2Z1( ) Si sp

Z1(

sp )

Z2(

sp )

= R2

cos θ tr cos θi

[7.222] T2

It is left to the reader as a short exercise to show that RP + TP = 1 , as should be since energy is conserved provided the two fluids are taken as a whole. Finally, a quantity commonly used is the transmission loss coefficient expressed in decibels as: TL = −10log10 TP

[7.223]

7.2.2.2 Transmission through a flexible wall: “infinite” and “finite” wall models In most applications it is also necessary to consider the case of fluids separated by flexible walls. This is typically the case in room’s acoustics, where the room stands for an enclosure bounded by walls, panels, glass windows etc. Sound transmission through such obstacles is of major importance in practice to assess the acoustic isolation and reverberation properties of the room. Taking the example of a short transient such as a handclap, the sound heard in a large and empty enclosure bounded by reflecting walls is generally characterized by one or two individual echoes followed by a signal of much longer duration than the initial source, which is called reverberant field. Echoes are due to discrete reflections separated by moments of silence. The reverberant field results from the multiple reflections of sound by the enclosure boundary. Indeed, the distance travelled by the sound depends on the


653

number of reflections and the latter are unavoidably associated with some loss of sound energy due to absorption and transmission at the boundary. Therefore, the rate of arrival to the listener of the reflected signals rapidly increases as time elapses since the occurrence of the initial transient and their magnitude decreases. As a consequence, the individual echoes merge into a continuous component which grows progressively up to a maximum which occurs when the rate of energy absorbed by the walls equals that delivered by the source. After the maximum is reached, sound energy decays as an exponential according to a characteristic time called reverberation time, which depends on the rate of sound absorption and transmission at the walls. Relatively short reverberation times are desirable in a theatre to optimize speech intelligibility, whereas relatively long reverberation times are desirable in music performance halls because, quoting [OLS 67], “the prolongation and blending of musical tones due to reverberation produce a more pleasant music performance”. Specialized literature on the subject is particularly abundant; see for instance [PIE 91], [BAR 93], [CRE 82], [KUT 00], [FAH 01]. The introductive presentation given here is restricted to the essential and basic aspect of sound transmission through a solid stucture which can be understood as follows. Due to the presence of sources of sound, lying for instance within the room, the internal faces of the solid boundaries are excited by the fluctuating pressure field and vibrate, mainly, if not exclusively, in the normal direction. As a consequence, the vibrating structure acts in turn as an acoustic source emitting both in the fluids inside and outside the room. Quantitative analysis of the process is however generally made extremely difficult for several reasons related to the calculation of the structural vibration and to the sound radiation. To make the problem tractable, two contrasting cases are usually considered successively. The first one corresponds to the so-called short wave limit, which means that the sound wavelengths of interest are much shorter than the typical size of the wall. If such is the case, the structural vibration induced by the sound pressure field is also of the short wave type, which means that the support conditions of the wall tend to become unimportant and so the concept of natural modes of vibration. As a consequence, the vibroacoustic coupled system is described in terms of travelling waves while the wall is supposed to extend to infinity. This kind of analysis is illustrated in the next subsection taking the example of plane sound waves impinging on a stretched plate. It will be shown that provided the phase velocity of the structural wave is sufficiently high, a plane wave is excited in the fluid. The second case is that of a finite structure which can be described in terms of natural modes of vibration and which is immersed in an infinite extent of fluid. If such is the case, the vibroacoustic problem can be treated, in principle at least, by computing the response of the plate to the fluctuating pressure difference between the two opposite faces of the plate and the pressure field induced by the motion of the plate; which also is a coupled problem. As a preliminary, to help understanding the physical background of such approaches let us consider a window pane edge lengths Lx = Ly = 2 m , thickness h = 4 mm , density ρ s = 2300 kg/m 3 , Young’s modulus Es = 61010 Pa , Poisson’s

654


ratio ν s = 0.3 . The pane is assumed to be hinged on the four edges and pretstressed along the Ox and Oy directions by a tensile force F ( ) . Plate stretching is introduced here not to keep close to reality, but for theoretical convenience in modelling either flexural waves or membrane waves, depending on the relative importance of the elastic to the prestress stiffness operator. As shown for instance in [AXI 05] Chapter 6, the modal equation of the pane is: 0

⎛ ∂ 4Z ⎛ ∂ 2Z ∂ 2Z ⎞ ∂ 4Z ∂ 4Z ⎞ +2 2 2 + − F (0) ⎜ + − ω 2 ρ s hZ = 0 D⎜ 4 4 ⎟ 2 2 ⎟ ∂x∂y ∂y ⎠ ∂y ⎠ ⎝∂ x ⎝∂ x 2 ∂ Z ∂ 2Z = 0 ; longitudinal edges : =0 all edges : Z = 0 ; lateral edges : 2 ∂x ∂ y2

[7.224]

where D designates here the bending stiffness coefficient of the plate, given by: D=

Es h 3 12 1 − ν s2

(

[7.225]

)

As extreme cases, either the flexural term or the stretching force can be negligible. In the first case the pane is modelled as a membrane. Transverse waves are not dispersive and travel at the phase velocity cψ( ) = F ( ) / ρ s h . In the seond case, the m

0

pane is modelled as a bended plate. Flexure or bending waves are dispersive and travel at the phase velocity cψ( ) = ω1/ 2 ( D / ρ s h ) b

1/ 4

, provided at least that transverse

shear across the plate thickness remains negligible, which requires that the wavelengths of interest remain much larger than h. The normalized mode shapes are: ⎛ nπ x ⎞ ⎛ mπ y ⎞ ϕ n ,m ( x, y ) = sin ⎜ ⎟⎟ n, m = 1, 2, 3... ⎟sin ⎜⎜ ⎝ Lx ⎠ ⎝ Ly ⎠

[7.226]

The related natural pulsations are: 1

ωn , m

2 2 2 ⎫⎞2 ⎛ 2 2 ⎧ ⎛ ⎞ ⎛ ⎛ ⎞ ⎛ mπ ⎞ ⎞ ⎪ ⎟ ⎛ ⎞ π π 1 n m ⎪ ⎜ (0) ⎜ ⎛ nπ ⎞ ⎜ ⎟ =⎜ +F ⎟⎟ ⎟⎟ ⎟ ⎬ ⎟ ⎨D ⎜ ⎟ + ⎜⎜ ⎜ ⎟ + ⎜⎜ ρ h L L L L ⎜ ⎟ ⎜ s x y x y ⎝ ⎠ ⎝ ⎠ ⎪ ⎜ ⎝ ⎠ ⎠ ⎝ ⎠ ⎟⎠ ⎪ ⎟ ⎝ ⎩ ⎝ ⎭⎠ ⎝

[7.227]

In Figure 7.45, the fifty first natural frequencies of a pane are plotted; the lefthand diagram refers to the case of bending without membrane stiffness in contrast with the right-hand diagram which accounts for bending and no membrane stiffness. In both cases, the modal wavelengths are λn ,m = 2 L / n in the Ox direction and


655

Figure 7.45. Natural frequencies of the flexure modes (left-hand plot) and membrane modes (right-hand plot) of the window pane

λn ,m = 2 L / m in the Oy direction. Hence shortest waves considered in the present

example are λ50,50 = 8 cm . Nevertheless, a difference of major importance is also made conspicuous in these diagrams, which concern the modal density. It can be noted that to cover the whole audio range, in the case of bending it is necessary to retain less than two thousand modes whereas about sixteen millions of modes would be necessary in the case of membrane modes. As already mentioned in Chapter 5 subsection 5.1.2.1 for so high values of modal density, the modal approach to the problems becomes unsuitable. Furthermore, even the concept of natural mode of vibration can be rightly questioned based on physical reasoning. In the present example, to analyse acoustic transmission through the membrane by using the modal approach, would imply to account for wavelengths as short as about one millimetre. Not only validity of the membrane model could be rightly questioned at such a length scale but, even more important, the very existence of standing waves is highly unrealistic because of the unavoidable presence of damping. To solidify this important point suffices to consider a damped travelling wave and to convert the time attenuation of the wave amplitude into space attenuation from the source. This means to express the amplitude of the damped harmonic wave at time t and distance x from the source as: Z ( x; t ) = Z 0 e −ως t e

iω ( t − x / cψ )

⇒ Z ( x = cψ t ) = Z 0 e −ς kx

[7.228]

Therefore, even if damping ratio is as small as 10−3 , amplitude of the travelling wave λ = 1 mm is devided by about 500 after a distance of one meter! Therefore the vibroacoustic problem will be analysed in subsections 7.2.2.3 and 7.2.2.4 based on the membrane model and travelling waves, while in subsection 7.2.2.5 it will be treated based on the thin plate model and associated structural modes of vibration. Another important point concerning the forced response of a plate to a pressure wave, is the fact that even if the incident sound wave is plane, the deflection of the plate is not necessarily uniform and therefore the sound waves it emits are not necessarily plane. This computational difficulty can be demonstrated by analysing

656


the plate response to a plane and harmonic pressure wave impinging at normal incidence on the plate. Using the modal relations [7.226] and [7.227], the plate response is found to be:

Z ( x, y ; ω ) =

4 ρ e hLx Ly

∞

∑ n =1

⎛ nπ x ⎞ ⎛ mπ y ⎞ sin ⎜ ⎟⎟ Pn ,m ⎟ sin ⎜⎜ ⎝ Lx ⎠ ⎝ Ly ⎠ ∑ 2 2 m =1 ωn , m − ω + 2iωωn , mς n , m ∞

[7.229]

where ρ e is the equivalent density of the plate vibrating within the fluid and Pn ,m is the modal projection of the exciting pressure field. In the particular case of a plane wave at normal incidence calculation of Pn ,m is immediate: L

L

Pn ,m =

⌠ x ⎮ ⎛ P0 ⎮⎮ sin ⎜ ⎝ ⎮ ⎮ ⌡0

nπ x ⎞ ⎟ dx Lx ⎠

⌠ y ⎮ ⎮ ⎮ ⎮ ⎮ ⌡0

(

)(

P0 Lx Ly 1 − ( −1) 1 − ( −1) ⎛ mπ y ⎞ sin ⎜ dy = ⎜ L ⎟⎟ nmπ 2 ⎝ y ⎠ n

m

)

[7.230]

Figure 7.46. Sound transmission loss for an infinite glass pane

At this step, it is useful to notice that the computed response Z ( x, y; ω ) behaves differently according to the order of truncature of the series [7.229] in relation with the circular frequency of the exciting wave. As could be anticipated, and illustrated in the right-hand side plot of Figure 7.46, if the excitation frequency lies within the frequency range of the vibration modes retained in the series, even if the pressure loading is uniform, the response of the plate is not. The shape of the deflected plate is highly sensitive to the nearly resonant modes and to the associated generalized forces Pn ,m . Accordingly, even if the exciting pressure wave is plane, the transmitted one is not. In contrast, if ω is higher than the highest modal frequency retained in the model, the computed response lies within the quasi-inertial range. In this spectral domain, the modal stiffness coefficients, hence the natural frequencies, can be neglected to compute the series [7.229] and correlatively the resonant character of the response vanishes. Moreover, it can be noted that if the natural frequencies ωn ,m


657

are set to zero, the modal expansion [7.229] reduces to the double Fourier series of a two dimensional rectangular pulse of length Lx and Ly , respectively. As illustrated in the left-hand side plot of Figure 7.46, where the first 40x40 modes are retained in the series, whatever the excitation frequency may be, if ωn ,m are set to zero, the computed response is essentially flat over the whole plate, except the inevitable Gibbs oscillations at the boundaries. Therefore, the emitted sound wave is also a plane pressure wave propagating in the same normal direction as the incident wave. Such a result can be extended to oblique incidence as verified later in subsection 7.2.2.4. Therefore, the “infinite plate” or short wave limit can be understood not only in terms of travelling waves but also in terms of standing waves as the limit case of a plate whose dimensions are so large with respect to the wavelength of the incident pressure wave that the natural frequencies are negligibly small while sufficiently short wave modes are to be retained into the modal basis. 7.2.2.3 Vibroaoustic travelling waves in an “infinite” membrane The free vibrations of the vibroacoustic system are governed by the following system of equations: ⎛ ∂ 2Z ∂ 2Z ⎞ 2 − F (0) ⎜ + ⎟ − ω ρ s hZ + p = 0 2 ∂ y2 ⎠ ⎝∂ x ∂ 2 p ∂ 2 p ∂ 2 p ω2 + + + p − ρ f ω 2 Zδ ( z ) = 0 ∂ x 2 ∂ y 2 ∂ z 2 c 2f

[7.231]

where the membrane is loaded by the pressure field p defined as the difference of pressure between the two opposite faces of the membrane and the fluid is loaded by the volume velocity source induced by the transverse motion Z of the membrane. Due to the antisymmetry of the fluid problem about the midplane of the membrane, solution is restricted to one half-space, for instance the domain z ≥ 0 . Then, we search for nontrivial waves related to a transverse vibration of the membrane of the harmonic and travelling type: Z = Z0e

i (ω t − k x x − k y y )

[7.232]

which stands for an outgoing wave travelling in the x and y positive directions, provided the wave numbers k x and k y are positive. To fulfil the coupling terms, the related pressure field must be of the form: p = P0 e

i (ω t − k x x − k y y − k z z )

[7.233]

which represents a pressure outgoing wave travelling in the z ≥ 0 domain, provided all the wave numbers k x , k y and k z are positive. Once again, substituting the wave [7.232] and [7.233] into the coupled system [7.231] one obtains a system of two dispersive equations which are solved and then discussed in relation to the complex or real nature of the roots and, in the last case, to their sign. The algebra can be

658


further alleviated by considering line waves travelling in one direction only, along Ox for instance. Therefore the system [7.231] is suitably rewritten as: pδ ( z ) ∂ 2Z − k s2 Z − =0 ∂ x2 F (0) ∂ 2p ∂ 2p + + k 2f p − ρ f c 2f k 2f Z δ ( z ) = 0 ∂ x2 ∂ z2

[7.234]

where: k s = ω / cs = ω F ( ) / ρ s h 0

; k f = ω / cf

[7.235]

Substitution of the waves [7.232] and [7.233] into the pressure equation of system [7.234] results in the dispersive equation: k z2 = k 2f − k x2

[7.236]

The mathematically simple relation [7.236] is also remarkable for its physical content since it implies that to obtain a plane sound wave travelling along an oblique direction within the half-space z > 0, k x must be less than k f . In other terms, the structural wave: cx > c f . Note that cx characterizes the phase speed of the membrane wave coupled to the fluid; that is precisely the reason why it differs from cs . Figure 7.47, built on the same basic principle as Figure 5.17, helps one to understand the condition cx > c f for radiation of a plane sound wave coupled to the membrane wave. Actually, the process is similar in the present transmission problem and in the guided wave problem treated in Chapter 5, since λ f and λx are related to each other by the same angular condition. Therefore it is expected that here also a cut-off frequency exists below which no travelling pressure wave can be induced. Pressure field is then adjusted to the fluid-structure interface condition, which gived the admissible pressure wave: p=

ρ f c 2f k 2f Z 0 2 x

k −k

2 f

e(

i ωt − k x x )

[7.237]

Substituting the displacement and pressure waves [7.232] and [7.237] into the membrane equation provides us with a second equation of dispersion which relates the waves numbers k x and k z to k s and k f . It is first written as: k x2 − k s2 +

ρ f c 2f k 2f F(

0)

k x2 − k 2f

=0

[7.238]


659

Figure 7.47. Membrane and associated pressure waves at a fixed time. The wavy line along the Ox direction stands for the Z profile of the membrane wave and the oblique lines for the traces in the plane Oxz of the planes of equal phase of the pressure wave in the z ≥ 0 half space. Traces in full lines and marked by (P) are for pressure peaks and the dashed line marked by (T) are for pressure troughs. Direction of propagation of the plane sound wave makes the angle θ with the transverse Oz direction

After a few standard manipulations, equation [7.238] can be further transformed into the following cubic equation in K = k x2 : K 3 − ( 2k s2 + k 2f ) K 2 + k s2 ( 2k 2f + k S2 ) K + μ 2f γ 4 k 4f − k s4 k 2f = 0

[7.239]

By using relation [7.235] which defines the speed of the membrane wave in vacuum to transform the last term in [7.238], one obtains: ρ f c 2f F

( 0)

=

ρ f c 2f ρ s hcs2

= μfγ 2

; μf =

ρf ρs

; γ =

cf cs h

[7.240]

A wave travelling along the direction Ox is necessarily related to a real and positive wave number k x . Therefore interest is restricted here to the roots of equation [7.239] which are real and positive. As the coefficients of the cubic are real, depending on their numerical values, a single root, or all the roots are real. Furthermore, remembering that to obtain a pressure wave travelling to infinity in the 0z direction k x must be less than k f , as a possible occurrence, K can be positive while k x is larger than k f , which describes a surface wave propagating along the membrane and confined in the transverse direction.

660


Figure 7.48. Membrane moderately stretched in air at STP

Figure 7.49. Membrane highly stretched in air at STP. Frequency domains: (1) no wave, (2) a single vibroacoustic wave, (3) three distinct vibroacoustic waves

If K is positive and k x less than k f , the solution stands for a vibroacoustic wave travelling along the Ox direction at the plate surface and within the fluid as a wedge shaped wave which clings to the membrane wave. Half angle of the wedge is θ defined by relation: ⎛k θ = sin −1 ⎜ x ⎜k ⎝ f

⎞ ⎟⎟ ⎠

[7.241]

Existence of such vibroacoustic travelling waves is illustrated in Figures 7.48 and 7.49 for two cases of academical interest based on the plane geometry already specified in subsection 7.2.2.2. In this example if the membrane tensioning is too low, there exists no travelling wave. Then, in an intermediate range of tensioning, a single vibroaoustic wave occurs above a certain cut-off frequency. Finally, three distinct vibroaoustic waves occurs if F0 and f are high enough. Note that in the present example, the domain where three distinct waves can coexist corresponds to very high and unrealistic values of tension. There is no place to present here a thorough discussion of the features of the vibroacoustic waves in relation with their frequency and the material and structural


661

properties of the fluid-structure coupled system. For more information, the reader can be reported to [MOR 86] where the case of a stretched wire and that of a flexed plate are discussed in depth. 7.2.2.4 Sound transmission through an “infinite” membrane, or plate Let us consider a plane harmonic pressure wave denoted pi which impinges on a membrane, or plate, at a given angle of incidence θ. This wave is reflected from the plate and let it vibrate. In turn, the plate, which is assumed to extend to infinity, acts as a sound source which emits a forced acoustic wave in both fluid half-spaces, that of the incident and reflected wave on one side, and that of the transmitted wave, on the other. The problem is sketched in Figure 7.50 which specifies the coordinate system used. The complex amplitudes of the forced plate displacement and pressures in the acoustic waves are assumed to be of the type: Z ( x; ω t ) = Z 0 e (

i ωt − k x x )

pi ( x; ω t ) = Pe i

i (ω t + k f ( z cosθ − x sin θ ) )

pr ( x; ω t ) = Pr e pt ( x; ω t ) = Pe t

i (ω t + k f ( − z cosθ − x sin θ ) )

[7.242]

i (ω t + k f ( z cos θ − x sin θ ) )

where pi designates the incident pressure wave, pr the reflected component and pt the transmitted component. In agreement with the remark made above concerning the shape of the plate response in the quasi-inertial range, the plate motion is assumed to force plane sound waves.

Figure 7.50. Sound transmission through a vibrating membrane practically infinite, separating a same fluid extending also to infinity

662


Coefficient of acoustic transmission of the membrane is suitably defined as the dimensionless ratio: TP =

pt

2

pi

2

[7.243]

Since the fluid is assumed to be the same in the half-space of incidence as in the half-space of transmission, the coefficient [7.243] can be directly interpreted as a ratio of radiated power. As could be anticipated, in practice it is preferred to express this ratio in a logarithmic scale by defining the so-called transmission loss coefficient as: RTL = −10log10 TP

[7.244]

TP can be determined by using the forced membrane and sound equations, as follows. The damped plate equation reads as:

D

2 ∂ 4Z (0) ∂ Z − F + Cs Z + ρ s hZ = pt − ( pi + pr ) ∂ x4 ∂ x2

[7.245]

Substituting the field components [7.242] into [7.245] and using the appropriate quantites already defined in the last subsection, one obtains: ⎧ ⎫ ⎛ ⎛ c( m ) ⎞2 ⎛ c (b) ⎞4 ⎞ ⎪ ⎪ − ik x sin θ ψ ψ ⎟ +⎜ ⎟ − 1⎟ ω 2 + iωCs ⎬ Z 0 e − ik x x = ( Pt − Pi + Pr ) e f ⎨ ρ s h ⎜ ⎜⎜ ⎟ ⎜ ⎟ ⎜ c ⎟ c ⎝⎝ x ⎠ ⎝ x ⎠ ⎠ ⎩⎪ ⎭⎪

[7.246]

As in the case of the free wave problem, the same angular relation [7.241] is recovered; which sets here the phase speed along the Ox direction to the supersonic value cx = c f / sin θ . Turning now to the coupling terms entering into the pressure wave equation, they imply: ∂ ( pi + pr ) = ik f cos θ ( Pi − Pr ) = ω 2 ρ f Z 0 ∂z + z =0 ∂pt ∂z

[7.247]

2

z = 0−

= ik f cos θ Pt = ω ρ f Z 0

As a corollary, the pressure components verify the two following conditions: Pt = Pi − Pr

; iω Z 0 =

− Pt cos θ ρ f cf

[7.248]

Using the results [7.248] to eliminate the reflected pressure component in equation [7.246], one obtains:


⎛ ⎛ c ( m ) ⎞2 ⎛ c (b) ⎞4 ⎞ ⎛P ⎞ρ c −iωρ s h ⎜ ⎜ ψ ⎟ + ⎜ ψ ⎟ − 1⎟ + Cs = 2 ⎜ i − 1⎟ f f ⎜ ⎟ ⎜ ⎟ ⎜ c ⎟ c ⎝ Pt ⎠ cos θ ⎝⎝ x ⎠ ⎝ x ⎠ ⎠

663

[7.249]

which is finally transformed into the inverse form of [7.243] as: pi

2

pt

2

⎛ C cos θ = ⎜1 + s ⎜ 2ρ f c f ⎝

2 ⎞ ⎛⎜ ωρ S h cos θ ⎟⎟ + ⎜ ⎠ ⎝ 2ρ f c f

⎛ ⎛ c ( m ) sin θ ⎜⎜ ψ ⎜⎜ cf ⎝⎝

2

⎞ ⎛ cψ(b ) sin θ ⎟ +⎜ ⎟ ⎜ cf ⎠ ⎝

4 ⎞⎞ ⎞ ⎟ − 1⎟ ⎟ ⎟ ⎟⎟ ⎠ ⎠⎠

2

[7.250]

Since we deal here with forced waves, the angle of incidence θ and the circular frequency can be treated as independent variables. Therefore, the transmission loss factor depends on both θ and ω. Formula [7.250] predicts that at very low frequency, if structural damping Cs is negligible, the incident wave is practically entirely transmitted. However such a mathematical result is physically irrelevant because if frequency tends to zero, the wavelength tends to infinity, precluding thus the validity of the the “infinite plate” model. In an intermediate frequency range, the structural membrane and bending terms are also found to be much less than unity. For instance, in a glass pane 3 mm thick in b air, cψ( ) is less than c f in the frequency range below about 4 kHz. For realistic values of tensioning forces, cψ(

m)

is much smaller than cψ( ) . Therefore, at sufficiently b

low frequencies expression [7.250] simplifies into the broadly know mass attenuation law, where damping is also neglected: ⎛ p RTL = 10log ⎜ i ⎜ p ⎝ t

⎛ ⎛ ωρ h cos θ ⎞ ⎟ 10log ⎜ 1 + ⎜ s 2 ⎟ ⎜ ⎜⎝ 2 ρ f c f ⎠ ⎝

2

⎞ ⎟⎟ ⎠

2

⎞ ⎟ ⎟ ⎠

[7.251]

In air, and in the audiofrequency range, provided θ is not to close to π / 2 (grazing incidence) the argument of the logarithm can be simplified by neglecting the first term with respect to the second one. The oblique mass attenuation law is thus further simplified into: ⎛ ωρ h cos θ RTL = 20log10 ⎜ s ⎜ 2ρ c f f ⎝

⎞ ⎟⎟ ⎠

[7.252]

According to the formula [7.252], the transmission loss increases by 6dB if either the wave frequency or the mass per unit area of the plate is doubled. Furthermore, in a diffuse acoustic field characterized by a random and nearly uniform distribution of incidence angles, the law can be suitably averaged to produce an attenuation coefficient independent of the incidence angle, see for instance [FAH 01]. On the other hand, relation [7.250] shows that the pane can be practically transparent to the sound waves for certain combinations of frequency and angle of incidence, provided damping is small. Neglecting the membrane stiffness in [7.250] practically no transmission loss exists if the following relation is verified:

664


cψ( b ) sin θ cf

= 1 ⇔ cψ( ) = 2π f co 4 b

c D 1 = f ⇔ f co = ρ s h sin θ 2π

ρsh ⎛ c f ⎞ ⎜ ⎟ D ⎝ sin θ ⎠

2

[7.253]

Figure 7.51. Sound transmission loss for an infinite glass pane

For the given angle of incidence θ, at the frequeny f co the condition for resonant excitation of the free vibroacoustic wave is fullfiled and the plate presents practically no resistance to the incident wave which is thus fully transmitted through the plate, except a small part which is dissipated by the damping term. The resulting notched curves of transmission loss coefficient are illustrated in Figure 7.51 for a glass pane 12 mm thick and a few angles of incidence. It can be noted that the notched and poorly attenuated part of the curves lies within the audiofrequency range in a broad domain of incidence angles. At such frequencies amplitude of the pane vibration can be significant. 7.2.2.5 Transmission through a finite plate When either the fluid or the structure is finite in size, standing waves and resonances have to be accounted for, adding complication to an already intricate problem, which is generally not amenable to analytical calculation, even in the simplest geometries, unless simplifying approximations of questionable validity are assumed. Here we content ourself with outlining the essentials of the analytical formulation for the same example as above, except that now the dimensions of the plate are not very large if compared to the sound wavelengths of interest. The object of the presentation is strictly limited to point out a few salient features and difficulties encountered in this type of problems. The reader interested in a more advanced presentation can be reported to [MOR 86] where he will find a thorough analysis of examples, including that of a circular membrane.


665

Let pi be a plane pressure wave impinging on a rectangular plate of edge lengths Lx and Ly , which is supposed to be embedded in an infinite rigid and fixed baffle to separate the fluid into two distinct subspaces, as in the case of the “infinite” plate. The complex amplitude of pi is written as: pi = P0 e (

i ω t − k ( sin θ x + cosθ z ) )

[7.254]

where the wave is assumed to be tilted by the angle π / 2 − θ with respect to the plane of the plate and constant in the Oy direction to alleviate algebra. As the problem is linear, the superposition principle applies which allows one to express the pressure field loading the plate as the sum of two distinct components. The first component is the pressure resulting from the incident and reflected wave, assuming the plate is fixed. Therefore it is twice the incident pressure on the plate (z = 0). The second component is related to the pressure resulting from the fluid-structure coupling term. It is twice the pressure induced by the plate motion on one face, since at the time a face of the plate pushes on the fluid within the incident half-space the other face pulls on the fluid within the transmittedt half-space. Whence the following formula: ⌠

Lx

P( x, y ,0) = 2 pi + 2 ρ f ω 2 ⎮⎮ dx0 ⌡0

L

⌠ y ⎮ ⎮ ⌡0

G ( x − x0 , y − y0 , 0 )Z ( x0 , y0 ) dy0

[7.255]

where the fluid-structure component is formulated using the Kirchhoff-Helmholtz integral, like in the case of the baffled piston. It is recalled that the Green function is of the type: G ( x − x0 , y − y0 , z − z0 ) =

e − ikr 2π r

; r=

( x − x0 )

2

+ ( y − y0 ) + ( z − z0 ) 2

2

[7.256]

Response of the plate to the loading [7.255] is marked by resonances which appear by expanding the response in a modal series. Assuming again hinged supports at four edges, the response is given by the series [7.229], where the generalized force is of the kind: L

L

Pn ,m =

⌠ x ⎮ ⎮ ⎮ ⎮ ⎮ ⌡0

⎛ nπ x ⎞ P ( x, y, 0 ) sin ⎜ ⎟ dx ⎝ Lx ⎠

⌠ y ⎮ ⎮ ⎮ ⎮ ⎮ ⌡0

⎛ mπ y ⎞ sin ⎜ dy ⎜ L ⎟⎟ ⎝ y ⎠

[7.257]

Already at this step it can be anticipated that response of the plate and sound emission by it has a resonant character when the frequency of the incident wave concides with a natural frequency of the plate, provided the corresponding generalized force [7.257] is not negligible. To proceed in the analysis, it is precisely necessary to calculate the Pn ,m coefficients and then the response of the plate. As could have been anticipated, this turns out to be a coupled problem because of the presence of the pressure field, denoted p fS , which is induced by fluid-structure

666


interaction. To make the problem tractable analytically, a simplified treatement of this term is necessary. The most drastic approximation possible consists in discarding the contribution of p fS to the plate response, arguing that magnitude of p fS is much less than that of pi ; which may be true, or not, depending on the

circumstances. In fact, as already illustrated by the one-dimensional problem analysed in Chapter 6 subsection 6.2.2.3, the behaviour of a vibroacoustic coupled system highly depends on the vibroacoustic damping ratio ς va . In agreement with the pressure field [6.95], the fluid-structure component can be neglected or not, depending whether ς va is a small quantity, or not. According to relation [6.92], ς va is likely to be small if the plate vibrates in a light gas such as air because of the small value of the fluid to structure mass ratio. Assuming this is the case here, the generalized force is written as: L

L

Pn ,m =

⌠ x ⎮ ⎛ 2 P0 ⎮⎮ sin ⎜ ⎮ ⎝ ⌡0

nπ x ⎞ − ik sin θ x dx ⎟e Lx ⎠

⌠ y ⎮ ⎮ ⎮ ⎮ ⎮ ⌡0

⎛ mπ y ⎞ sin ⎜ dy ⎜ Ly ⎟⎟ ⎝ ⎠

[7.258]

With the aid of formula [6.103], one obtains: Pn ,m =

2 P0 Lx Ly 1 − ( −1)

(

nmπ

2

m

) (1 − ( −1) e n

(1 − a ) 2 n

− ian nπ

)

; an =

kLx sin θ nπ

[7.259]

The important feature highlighted by formula [7.259] is the presence of resonant peaks of excitation at the space coincident wavelengths defined by the condition: an =

kLx sin θ 2 L sin θ =1⇔ λ = x nπ n

[7.260]

where λ = 2π c f / ω is the wavelength of the incident acoustic wave. Based on the results established in the former subsection, condition for space coincidence turns out to be of the same nature, whether the plate is modelled as an infinite or a finite solid. In both cases, space coincidence occurs if the structural wavelength matches to the sound wavelength, once it is projected onto the direction of sound propagation. The last step is to determine the transmitted wave induced by the vibration of the plate. The problem is suitably formulated by using the Kirchhoff-Hemholtz integral as: ⌠

Lx

pt ( x, y , z ) = ρ f ω 2 ⎮⎮ dx0 ⌡0

L

⌠ y ⎮ ⎮ ⌡0

G ( x − x0 , y − y0 , z )Z ( x0 , y0 ) dy0

[7.261]


667

Figure 7.52. Far pressure field induced by the vibrating plate in the transmitted wave halfspace z ≥ 0

Calculation can be performed in the farfield approximation, see Figure 7.52 which sketches the geometry of the problem. The pressure radiated into the transmitted wave half-space z ≥ 0 is obtained by summing the individual contributions associated with the natural modes of vibration of the plate. Using the Green function [7.256], the pressure related to the n,m mode is: L

(t )

pn ,m ( x, y , z , ω ) = ρ f ω

2

⌠ x ⎮ ⎛ An ,m ⎮⎮ sin ⎜ ⎝ ⎮ ⎮ ⌡0

nπ x0 ⎞ ⎟ dx0 Lx ⎠

L

⌠ y ⎮ ⎮ ⎮ ⎮ ⌡0

⎛ mπ y0 ⎞ e − ikr sin ⎜ dy ⎜ L ⎟⎟ 0 2π r ⎝ y ⎠

[7.262]

where the modal coefficient of participation An ,m is obtained from [7.229] and [7.259] as: An ,m =

(

)(

2 P0 1 − ( −1) e − ian nπ 1 − ( −1)

nmπ ρ e h (ω 2

n

2 n ,m

m

)

; an =

− ω + 2iωωn ,mς n ,m )(1 − an2 ) 2

kLx sin θ nπ

[7.263]

The far field approximation consists of assuming that the distance between the receiving point, denoted R, and any emitting point on the plate, denoted S0 , can be approximated by the distance r of R from the origin of the axis. Accordingly, relation [7.262] is transformed into: ⌠

pn( ,m) ( x , y , z, ω ) = ρ f ω 2 An ,m t

Lx

− ikr ⎮

⎛ e ⎮ sin ⎜ 2π r ⎮⎮⎮ ⎝ ⌡0

L

⌠ y ⎮ nπ x0 ⎞ ⎮ ⎟ dx0 ⎮ Lx ⎠ ⎮ ⎮ ⌡0

⎛ mπ y0 ⎞ − ik sinϕ ( x0 cosθ + y0 sin θ ) sin ⎜ e dy0 ⎜ L ⎟⎟ ⎝ y ⎠

[7.264]

668


Remaining integrals are of the same type as in [7.258]. Thus integration proceeds in a similar way, which brings in evidence new possible space coincidences leading to marked reinforcement in the transmitted wave amplitude. They depend on the angular position of the receiving point. 7.2.3

Radiation of water waves

Oscillations of floating bodies were addressed in Chapter 3 subsection 3.5.2, where the reader was warned that the sole purpose of the brief and elementary description given in this book is to provide the reader with a few basic ideas on this highly specialized and difficult subject without entering into the analytical and numerical techniques which are beyond the purview of the present book and also beyond the knowledge of the authors; this also holds for the considerations presented here. Radiation damping associated with gravity waves can be very significant for the oscillations of floating objects since the corresponding dissipative forces are often of the same order of magnitude as the fluctuating buoyancy or inertia forces. As a consequence, its importance is sensitive to the shape of the floating object and to the type of mode considered. So far as the solid is concerned, it is sufficient to restrict the problem to a single mode of the rigid body. Here, it is found convenient to select the heave mode of the cylindrical buoy already described in Chapter 3 subsection 3.5.2.5, see Figure 7.53. Actually, it may be anticipated that the heave mode is much more heavily damped than the rolling mode since the latter displaces no fluid, except a thin viscous boundary layer as further explained in section 7.3.

Figure 7.53. Heave mode of a floating circular cylindrical rod (strip model)

7.2.3.1 Energy considerations It is of interest to look at the problem from the energy standpoint. Referring to Figure 7.53, whatever the value of the spring constant K s may be, with the aid of


669

the stiffness coefficient [3.250] related to buoyancy, the mechanical energy of the oscillator can be written as: Em =

(K

s

+ 2 ρ f g ) Z 02

[7.265]

2

Of course, the cylinder vibrating according to a heave mode displaces a certain amount of fluid near the water level and thus triggers water waves which are never reflected back to the buoy. We are interested in calculating the radiation damping ratio. Assuming we are in deep water and denoting W0 the crest displacement of the outgoing waves, the latter can be written in terms of pressure as: p+ ( x, z; t ) = ρ f gW0 e kz e (

i ω0 t − kx )

kz i (ω0t + kx )

p− ( x, z; t ) = ρ f gW0 e e

x≥R ; z≤0 x ≤ −R ; z ≤ 0

[7.266]

Calculation of the energy radiated per cycle of vibration is immediate using a few results already established in Chapter 3 (formulas [3.25], [3.33] and [3.78]). The result is: ⎛ gW0 ⎞ ER = πρ f ⎜ ⎟ ⎝ ω0 ⎠

2

[7.267]

The energy loss ratio takes on the condensed and instructive following form: 2

2

⎛ g ⎞ ⎛ W0 ⎞ ⎛ W0 ⎞ ER = 2⎜ ⎟ = 2⎜ ⎟ 2 ⎟ ⎜ R ω Z Em ⎝ 0⎠ ⎝ 0⎠ ⎝ F Z0 ⎠

2

[7.268]

Where the pertinent Froude number is F = Rω02 / g . Incidentally, in deep water it can be also interpreted as a reduced wave number since F = Rω02 / g = kR . Of course, the crucial point to assess the amount of radiation damping is to determine the relative amplitude of the fluid to the structural displacement for a free oscillation, triggered either by an impulse or an initial vertical displacement of the solid with respect to the buoyancy centre. This cannot be achieved without solving the boundary value problem specified in the next subsection. However, as a preliminary, the two following points are of interest. First, relation [7.268] offers a possibility to determine radiation damping experimentally by measuring Z 0 and W0 . Actually, the method is not very attractive in three dimensions, but has been frequently used in two dimensional experiments, as mentioned in particular in [WEH 71]. The second point is a mere consequence of the fact that W0 can be expected to be a substantial fraction of Z 0 in some intermediate range of F values, leading to rather heavily damped heave oscillator. Such a trend can be illustrated using the computed data published in [DAM 00], which are reproduced in Figure 7.54 as indicative values. The plot on the left-hand side uses the original form

670


of data reduction used in the paper. The plot on the right-hand side refers to a cylinder of radius R = 10 cm floating on water. The reduced damping coefficient is plotted versus the angular frequency ω.

Figure 7.54. Radiation damping coefficients taken from [DAM 00]

Using these data, we are in position to plot the squared modulus of the transfer function H of the floating cylinder and to assess the equivalent viscous damping ratio, based on the half-power bandwidth method. Referring to equation [3.247], H is defined as:

(

H (ω ) = K w + iωCw (ω ) − ω 2 ( M a + M w (ω ) )

)

−1

[7.269]

The result is shown in reduced form in Figure 7.55, using the undamped resonance frequency f 0 and the maximum of HH * as scaling factors to reduce the corresponding quantities. As indicated in Figure 7.55, the resonance frequency is about 1.4 Hz, which corresponds to f r = F 0.8 and the equivalent viscous damping ratio is about 25%. 7.2.3.2 Boundary value problem Using the concepts and methods described in the context of radiation of sound waves in Chapter 5, we are in position to outline the basic principles of the integral formulation of the fluid part of the coupled problem. As in the system of equations [3.246], the heave mode is first described by the forced equation:

(K

− ω02 M s ) Z 0 = R ⎮⎮ ⌠

s

π /2

⌡− π / 2

p cos θ dθ

[7.270]


671

Figure 7.55. Squared modulus of the damped transfer function [7.269] near resonance

The right-hand side term of equation [7.270] stands for the harmonic pressure force exerted per unit length of the buoy. According to the theoretical methodology of general use in naval and ocean engineering (see for instance, [NEW 85], [MEI 89], [FAL 90], [FAL 02]), the fluid part of the problem is formulated in terms of the velocity potential defined by relation [1.51]. Once linearized about the steady free surface, the unsteady Bernoulli equation [1.77] implies: p = −iωρ f Φ − ρ f gZ 0

[7.271]

where Φ is the complex amplitude of the velocity potential. The oscillator equation [7.270] is thus transformed into:

(K

+ 2 ρ f g − ω02 M s ) Z 0 = −iωρ f R ⎮ ⌠

s

π /2

⌡− π / 2

Φ cos θ dθ

[7.272]

It is not difficult to show that Φ is governed by the following boundary value problem:

(K

+ 2 ρ f g − ω02 M s ) Z 0 = −iωρ f R ⎮ ⌠

s

π /2

⌡− π / 2

Φ cos θ dθ

ΔΦ = 0 grad Φ.n (W ) = iωρ f Z 0 cos θ

r=R ; −

⎛ 2 ∂Φ ⎞ ∂Φ =0 ; ⎜ −ω Φ + g ⎟ ∂ z ⎠ z=H ; x ≥R ∂z ⎝

π π ≤θ ≤ 2 2

[7.273]

=0 z =− H

Furthermore, in the far field approximation Φ must stand for an outgoing wave, written in the deep water case as:

672


Φ+

W0 + kz i (ωt −kx ) e e iω

x →∞

;

Φ+

x →−∞

W0- kz i (ωt + kx ) e e iω

; k=

ω2 g

[7.274]

Incidentally, it is noted that, as in the case of vibroacoustic coupling, radiation damping depends entirely on the far field approximation [7.274]; which is obviously not the case of the “added mass” effect. As already mentioned in Chapter 3, the most general and efficient method to solve numerically a boundary value problem of the type [7.273] is based on a boundary integral formulation of the radiation and scattering problem of the type [7.273]. Since Φ is solution of a Laplace equation, it can be checked that it also verifies an integral equation of the same type as the Kirchhoff-Helmholtz equation, which holds for the sound waves, and for incompressible fluids provided the asymptotic form of the Green function for c f tending to infinity is adopted. The boundary integral equation is of the generic type: ⌠ Φ ( r ; ω ) = ⎮⎮

⎮ ⌡(S )

(GgradΦ − ΦgradG ) .n dS

[7.275]

In equation [7.275], Φ designates the Fourier transform of the velocity potential and G that of an appropriate Green function. As shown in Figure 7.56, the closed surface (S ) comprises the wetted wall of the body (W ) , the sea floor (SH ) , the

free surface ( Σ 0 ) and finally a far field surface (S∞ ) within the liquid located at a large distance from the floating body. In the present example, the latter is composed of two vertical planes perpendicular to the Ox axis set at x±∞ >> R . In the case of a 3D problem, a vertical circular cylinder of radius R∞ >> R is selected as an

appropriate (S∞ ) boundary.

Figure 7.56. Domain of the boundary value problem

As already mentioned, a first possible choice for the Green function is the asymptotic form of the acoustic Green function in the limit of incompressible fluid. Hence, in the present context, equation [5.302] written in the frequency domain becomes:


⌠

⌠ ⎮ ⎛ ⎛ 1 ⎞ ⎞ ⎮ ⎮ 4π Φ ( r ; ω ) = ⎮ Φ ⎜ grad ⎜ ⎟ ⎟ .nd S − ⎮ ⎮ ⎝ r ⎠⎠ ⎮ ⎝ ⎮ ⌡

(S )

⌡(S )

1 gradΦ.nd S r

673

[7.276]

r is the distance from the source point lying on the closed surface (S ) to the

receiving point. If the latter is on (S ) the multiplying factor 4π is replaced by 2π. Clearly, the Green function defined just above, which is known as the Rankine source, verifies Laplace’s equation. However, it does not comply with any of the boundary conditions which hold at ( Σ 0 ) , (SH ) and (S∞ ) . More refined Green functions can be built which verify at least some of these conditions, yet at the cost of a significant increase in the analytical complexity, as detailed for instance in [THO 53], [WEH 60], [YEU 82], [NEW 85]. On the other hand, to deal with seakeeping problems, it is generally suitable to decompose Φ as the sum of three distinct velocity potentials: Φ = Φ I + Φ S + Φ IS

[7.277]

Here Φ I stands for a given incident wave which excites the solid body, Φ S describes the wave scattered or diffracted by the solid body which is treated as a rigid and fixed reflecting obstacle. Finally, Φ IS stands for the wave induced by the fluid-structure interaction mechanism. The conditions fulfilled by these individual components at the wetted wall are: [7.278] grad ( Φ I + Φ S ) .n = 0 ; grad ( Φ IS ) .n = i ω X s .n (W )

(W )

(W )

If Φ I vanishes so does Φ S . This is the case in particular if the body is excited by an initial impulsion or displacement. On the other hand, if the body is rigid and fixed, Φ IS vanishes. 7.3. Dissipation induced by viscosity of the fluid 7.3.1

Viscous shear waves

As already established in Chapter 1, subsection 1.3.3.5, the set of equations which govern the small vibrations of an incompressible and viscous fluid can be written as: divX f = 0 [7.279] ρ f X f + grad p − μ f ΔX f = 0 Here μ f designates the dynamic viscosity coefficient of the fluid. Equations [7.279] can also be transformed as:

674


Δp = 0 ΔX f − ν f Δ 2 X f = 0

[7.280]

where ν f is the kinematic viscosity coefficient. According to the first equation [7.280], p is governed by the incompressible law of an inviscid fluid and X f is governed by a viscous wave equation as evidenced a little later in this subsection. In such equations pressure and fluid displacement appear as two uncoupled fields, which can however be coupled through the no-slip condition to be fullfiled at a solid wall; the latter is expressed as: X f − X s =0 [7.281] (P )

To demonstrate that the second equation of system [7.280] can be interpreted as a wave equation and to highlight the major properties of such waves, it is found convenient to consider the problem sketched in Figure 7.57, which is known as the second Stokes problem. A horizontal layer of incompressible and viscous fluid is bounded at z = H by a free level where gravity effect is neglected and at z = 0 by a rigid plane wall which is assumed to vibrate in the horizontal direction Ox according to the prescribed harmonic law X 0 eiωt . As a particularly attractive feature of this problem, fluid motion is found to be independent of pressure and to be entirely governed by viscous shear. Actually, the displacement field is postulated to be bidimensional and of the harmonic form: X ( x, z, ω ) = ( Xi + Zk )eiω t [7.282] Nevertheless, it is easily realized that X and Z cannot depend on x, since the wall motion itself is independent of x. In other terms, the system remains identical for any horizontal translation transforming x into x+a, where a is an arbitrary length. Thus, fluid incompressibility implies that: ∂X ∂Z ∂Z [7.283] + = =0 divX f = ∂x ∂z ∂z Condition [7.283] states that to exist, a vertical component of fluid motion must be uniform. Moreover it is necessarily nil, because the wall has no vertical motion and the fluid sticks to the wall. Going a step further, by using the momentum equation it is shown that pressure is not a variable of the problem. Projection onto the horizontal and vertical axes of the vector equation [7.279], leads to: ⎧ 2 ∂ 2X −ω X − iων f =0 ⎪ ∂ z2 1 ⎪ −ω 2 X f + grad p − iων f ΔX f = 0 ⇒ ⎨ 1 ∂p ρf ⎪ =0 ⎪⎩ ρf ∂ z

[7.284]


675

Figure 7.57. Stokes second problem: viscous fluid layer excited by a prescribed tangential oscillation of a wall

Hence pressure is necessarily uniform and fixed to the nil value prescribed at the free level z = H: p ( z; ω ) = p ( H ; ω ) = 0

[7.285]

Therefore the system of equations [7.284] reduces to the horizontal momentum equation which is provided with the horizontal component of the condition [7.281] at the interface between the fluid and the solid: −ω 2 X − iων 0

∂ 2X =0 ∂ z2

[7.286]

X (0; ω ) = X 0

The general solution of differential equation [7.286] is readily found to be: X ( z ) = A+ e − k+ z + A− e − k− z ; k 2 =

iω iω ω ⇒ k± = ± = ±(1 + i ) νf νf 2ν f

[7.287]

Turning back to the complex amplitude of the harmonic motion, the incoming wave is clearly unphysical since it is found to grow as z increases. Hence the only physically meaningful solution is the outgoing wave: X ( z; ω t ) = De

−z

ω 2ν f

⎛

i ⎜⎜ ω t − z

e

⎜ ⎝

ω 2ν f

⎞ ⎟ ⎟ ⎟ ⎠

= X 0e

⎛ z − z i ⎜⎜ ω t −η ην ⎝ ν

e

⎞ ⎟ ⎟ ⎠

[7.288]

where it is found appropriate to define the characteristic viscous attenuation distance, which is frequency dependent, as: ην =

2ν f ω

[7.289]

676


Solution [7.288] describes a transverse and space evanescent wave since the fluid oscillates in the Ox direction whereas the wave travels along the Oz direction, with a magnitude which decays exponentially when z increases. This is a clear consequence of the dissipative nature of viscous forces. Concerning the characteristic lengh of decay, it is of interest to emphasize that in many cases of common occurrence ην is a very small quantity when compared to the smallest dimension of the fluid volume, that is H in the present example. For instance, in water η ν is about 200 µm for a vibration at 10 Hz. The classical result of fluid mechanics is recovered here; for poorly viscous fluids like water or air, in laminar flow regime, viscosity effects are concentrated essentially in the immediate vicinity of the wall within a thin layer of fluid, termed for that very reason the boundary layer. As an essential part of fluid mechanics, boundary layer theory is described in depth in many textbooks, let us mention in particular [SCH 79], [PAN 86], [FAB 01]. Here it suffices to recall that viscous shear necessary to accomodate the flow to the no-slip condition at the wall occurs within the boundary layer and so the energy dissipation. In the steady flow regime, the thickness of the boundary layer is controlled by the Reynolds number which measures the ratio of the inertial to the viscous forces within the fluid, based on a typical steady flow velocity and length scale. In oscillating and laminar flow regime, the thickness of the boundary layer is controlled by the oscillatory Reynolds number, or Stokes number, already defined in Chapter 1, formula [1.113] repeated here for convenience: Sν =

ωL2 νf

[7.290]

It may be noted that if ην / 2 is selected as the appropriate length scale in [7.290], a unit value for the Stokes number is obtained. Confinement of shear motion within the boundary layer is illustrated in the upper plot of Figure 5.58, where two particular profiles of fluid displacement are represented, namely X f ( z; ω t = 2nπ ) and X f ( z; ω t = (2n + 1)π ) . At such times the wall displacement is assumed to be zero ( X s = X 0 sin ω t ). Finally, it is worthwhile to notice that viscous shear waves are dispersive. Phase and group speeds are found to be: cψ = ωην = 2ων f

; cg = 2cψ

[7.291]

The lower plot of Figure 7.58 illustrates the wavy motion of the boundary layer duting a cycleof oscillation. It shows some marked similarity with the damped modal wave profiles depicted in Figure 7.14. In both cases, we have to deal with non standing waves associated with an energy outflow from a source, here the vibrating wall, to a sink, here the fluid layer.


677

Figure 7.58. Space profiles of the viscous shear wave. Upper plot: profiles taken at two times separated by half a period of the oscillating bottom plate and at which plate displacement is zero. Bottom plot: profiles taken at various times during one cycle

7.3.2

Fluid-structure coupling, incompressible case

7.3.2.1 Piston-fluid system The archetypical system already used many times in this book is revisited here in the particular case of an incompressible and viscous fluid. Fluid velocity and pressure field induced by the motion of the piston are assumed to be

678


Figure 7.59. Mass spring system coupled to a column of incompressible and viscous fluid

two-dimensional, apart from local 3D features taking place in the immediate vicinity of the piston. Discarding such local effects, the velocity field is written as: X f = W ( r )eiω t i ; p = p( x )eiω t [7.292] The system [7.280] reads here as: 2

d p =0 dx 2 1 dp −ν f iωW + ρ f dx p( L; ω ) = 0

;

⎛ d 2W 1 dW ⎞ ⎜ 2 + ⎟=0 r dr ⎠ ⎝ dr dp = ω 2ρ f X 0 ; dx x = 0

[7.293] W ( R) = 0

As an immediate consequence, pressure is the same as in the inviscid case: p ( x; ω ) = ω 2 ρ f X 0 ( L − x )

[7.294]

This result indicates the presence of the same component of inertia force exerted on the piston as in the inviscid fluid. Nevertheless, viscosity brings other force components into the system related to wall friction and to the modification of the fluid velocity field induced by fluid shearing. Accordingly the oscillator equation is written as: ⎡⎣ K s − ω 2 ( M s + M a )⎤⎦ X 0 = Fν

;

M a = M f = ρ f π R2 L

[7.295]

where Fν is the resultant viscous force exerted on the piston, which has to be determined. For that purpose, it is necessary to calculate first the velocity field, which is governed by the forced equation of motion: ω2 d 2W 1 dW iω + − = W X0 νf dr 2 r dr ν f

The solution is found to be:

; W ( R, ω ) = 0

[7.296]


⎛ ⎛ ω ⎞⎞ ⎜ J0 ⎜ r ⎟⎟ ⎜ iν f ⎟ ⎟ ⎜ ⎝ ⎠ W ( r ) = iω X 0 ⎜ 1 − ⎟ ⎜ J ⎛R ω ⎞⎟ ⎟ 0⎜ ⎜ ⎜ iν f ⎟⎠ ⎟ ⎝ ⎝ ⎠

679

[7.297]

The resultant of he shear viscous forces exerted at the tube wall follows as:

Fν = 2π RLμ f

dW dr

= iω X 0 2π RLρ f r=R

ων f i

⎛ ω R2 ⎞ J1 ⎜ ⎟ ⎜ iν f ⎟ ⎝ ⎠ ⎛ ω R2 ⎞ J0 ⎜ ⎟ ⎜ iν f ⎟ ⎝ ⎠

[7.298]

It is worthwhile noticing that the dimensionless argument in the Bessel functions is the Stokes number ω R 2 / ν f which refers to the tube radius as the scale length is concerned. Accordingly, it is interesting to discuss successively the two asymptotic cases of a large radius leading to a large value of the Stokes number and that of a small radius, leading to small values of the Stokes number. 1. Poorly confined fluid: ω R 2 / ν f >> 1 The Bessel functions are replaced by their asymptotic expansion restricted to the first order term: J n ( z)

2 cos ϕ n πz

⎛ 2n + 1 ⎞ ; ϕn z − ⎜ ⎟π ⎝ 4 ⎠

[7.299]

In the present application one obtains: ⎛ ω R2 ⎞ J1 ⎜ ⎟ ⎜ iν f ⎟ ⎝ ⎠ −i ⇒ F 2ω 2 M X (1 − i ) ν f ν 0 f 2 2ω R 2 ⎛ ωR ⎞ J0 ⎜ ⎟ ⎜ iν f ⎟ ⎝ ⎠

[7.300]

The real part of the viscous force means that viscosity indues another inertia component which has to be added to that which holds in inviscid case. It stems from the modification of the velocity field by viscous shearing, with respect to the non viscous case. It can be interpreted as an additional contribution to kinetic energy due to the oscillation of the fluid in the boundary layer. If the fluid is poorly confined, as it is assumed to be the case here, the added mass related to viscosity is negligibly small because the boundary layer is much smaller than R. On the other hand, the imaginary term in [7.300] is the dissipative component related to viscous friction. The oscillator equation [7.295] is thus written as:

680


⎡ ⎤ ων f − ω 2 ( M s + M f )⎥ X 0 = 0 ⎢ K s + 2iω M f 2 2R ⎣⎢ ⎦⎥

[7.301]

The above equation indicates that the friction force induced by fluid viscosity, which varies as ω 3/ 2 , does not identify to the so-called viscous damping model. Like radiation dissipative forces, viscous forces lag behind the structural displacement by an amount which depends on frequency. Nevertheless, as already illustrated in the case of radiation damping, if dissipation is small, an equivalent viscous damping ratio can be defined by fixing ω to the resonant value ω 1 . The equivalent viscous damping ratio reads here as: ςν =

Mf

νf

Ms + M f

2ω1 R 2

; ω1 =

K Ms + M f

[7.302]

Figure 7.60. Radial variation of the axial fluid velocity at distinct times expressed as a fraction of the period of oscillation (large value of Stokes number)

It is also instructive to look at the radial profile of the fluid velocity. Based on the asymptotic values [7.299], the velocity field [7.297] is expressed as: ⎛ ⎛ ω π⎞⎞ ⎜ cos ⎜ r − ⎟⎟ ⎜ iν f 4 ⎟ ⎟ ⎛ R ⎞⎜ ⎝ ⎠ W ( r ) iω X 0 ⎜ 1 − ⎟⎜ ⎟ r ⎠⎜ ⎛ ω π ⎞⎟ ⎝ ⎜ cos ⎜⎜ R iν − 4 ⎟⎟ ⎟ f ⎝ ⎠⎠ ⎝

[7.303]


681

Again, the profiles corresponding to the real oscillation can be deduced from [7.303] by retaining either the real, or the imaginary part of the complex quantity W ( r )eiω t , as shown in Figure 7.60 which is built in a similar way as Figure 7.14. Here the reduced axial velocity W / iω X 0 is plotted versus the reduced radial distance r/R. As expected, when the Stokes number is high, fluid velocity is essentially uniform in most part of the tube, except in a thin boundary layer near the tube wall where velocity is always zero. 2. Highly confined fluid: ω R 2 / ν f > h n

[7.305]

Figure 7.62. Fluid layer coupling two vibrating parallel plates

These geometrical particularities allow one to treat the problem by using a strip model where only the space variations along the Ox and the tranverse Oy directions are considered. Fluid motion is governed by the equations:


⎧ ⎪ ⎪ ⎪ ⎪⎪ ⎨iω v + 1 ⎪ ρf ⎪ ⎪ 1 ⎪iω u + ρ f ⎩⎪

683

V = vi + uj ∂v ∂u + =0 ∂x ∂ y ∂p −ν f ∂x

⎛ ∂2 v ∂2 v ⎞ ⎜ 2+ ⎟=0 ∂y2 ⎠ ⎝ ∂x

∂p −ν f ∂y

⎛ ∂2 u ∂2 u ⎞ ⎜ 2+ ⎟=0 ∂y2 ⎠ ⎝ ∂x

[7.306]

The boundary conditions read as: v ( x , − h ) = v ( x, h ) = 0 2

u( x, −h ) = −iωU n( ) sin kn x ; p ( 0, y ) = p ( Lx , y ) = 0

1

u( x, h ) = iωU n( ) sin kn x

[7.307]

We seek separated variable solutions of the type: v ( x, y ) = vn ( y ) cos kn x ; u ( x, y ) = un ( y ) sin kn x p ( x, y ) = pn ( y ) sin kn x

[7.308]

Furthermore, in agreement with the thin fluid layer approximation already introduced in Chapter 2 subsection 2.3.3 the pressure gradient in the transverse direction is neglected. So the pressure field is simplified into: p ( x ) = pn sin kn( ) x ν

[7.309]

Accordingly, the momentum equation projected onto the Ox direction is written as: iω vn +

⎛ d 2v ⎞ kn p − ν f ⎜ 2n − kn2 vn ⎟ = 0 ρf dy ⎝ ⎠

[7.310]

The solution is found to be: (ν ) y

(ν ) y

vn ( y ) = ae kn

+ be − kn

− αn

(k ( ) )

iω νf

αn =

ν n

2

= kn2 +

;

kn p

( ) (ν )

ρ f ν f kn

[7.311] 2

ν

where kn( ) designates the viscous modal wave number, which is a complex quantity. With the aid of the boundary conditions, the modal fluid velocity is found to be: vn ( y ) =

(

(

) ( ( ) cosh ( k h ) ν

ν

α n cosh kn( ) y − cosh k n( ) h ν n

))

[7.312]

684


Then transverse fluid velocity can be obtained by using the mass equation, which implies: ∂u ∂u ∂v =− ⇒ kn un ( y ) = n ∂y ∂x ∂y

[7.313]

Whence: un ( y ) =

( (

α n k n sinh kn( ) y − ykn( ) cos kn( ) h

) cosh ( k ( ) h ) ν

(

ν

ν

ν n

)) + C

[7.314]

where C stands for an undetermined constant. The conditions at the fluid-structure interfaces imply: iωU n( ) = ( 2)

iωU n =

(ν )

kn

kn(

(sinh ( k h ) − k h cosh ( k h ) + C ) cosh ( k h ) αk ( − sinh ( k h ) + k h cosh ( k h ) + C ) cosh ( k h ) α n kn

1

(ν ) n

(ν )

(ν ) n

(ν ) n

n

ν)

n n

(ν )

(ν )

n

(ν ) n

[7.315]

(ν )

n

n

and so:

(

iω U n( ) − U n( 1

2)

2α k sinh ( k ( ) h ) − k ( ) h cosh ( k ( ) h ) ) ) = k ( )h cosh ( ( ) k h ( ) ν n

n n

ν n

ν n

ν n

ν n

[7.316]

Fluctuating pressure follows as: pn = iωρ f ν f

pn = ρ f

(U

(U ( ) − U ( ) ) ⎛ k ( ) ⎞

(1) n

1 n

2

n

2h

(2)

− Un 2h

)

(

) (

⎛ kn(ν ) h cosh kn(ν ) h ⎜ ⎜⎜ ⎟⎟ (ν ) (ν ) (ν ) ⎝ kn ⎠ ⎜⎝ sinh kn h − kn h cosh kn h ν n

2

(

)

⎛ ⎜ ⎛ ω 2 L2x ⎞⎜ 1 ⎜ 2 2 − iων f ⎟ ⎜ ν tanh kn( ) h ⎝n π ⎠⎜ − 1 ⎜ ν kn( ) h ⎝

(

)

)

⎞ ⎟⇒ ⎟ ⎠

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

[7.317]

It is instructive to calculate the components of the Stokes stress tensor: σ xx = − p + 2 ρ f ν f

∂v ∂u ; σ yy = − p + 2 ρ f ν f ; σ xy = 2 ρ f ν f ∂x ∂y

⎛ ∂v ∂u ⎞ ⎜ + ⎟ ⎝ ∂y ∂x ⎠

[7.318]

At the walls, the no-slip condition implies that: ∂v ∂u =− ∂ x ( y=± h ) ∂y

=0 ( y= ± h )

[7.319]


685

Another interesting point to note is that the work performed by the longitudinal normal stress σ xx and by the viscous shear stress σ xy is nil since according to the Kirchhoff-Love model (cf. [AXI 05], Chapter 6), the plate vibration is purely transverse. As a consequence, the only stress component which produces some work is the transverse normal component σ yy . By virtue of relation [7.319], σ yy reduces to a pressure just like in the inviscid case. However, fluid viscosity still enters into the problem, as p depends on viscosity as made evident in [7.317]. The generalized force exerted by the fluid on a plate is found to be: L

Qn =

⌠ x ⎮ − pn ⎮⎮ ⎮ ⎮ ⌡0

2

⎛ ⎛ nπ x ⎞ ⎞ ⎜⎜ sin ⎜ L ⎟ ⎟⎟ dx ⇒ ⎝ ⎝ x ⎠⎠

⎛ ⎜ ⎞ ⎜ U n(1) − U n( 2 ) Lx ⎛ ⎛ ω Lx ⎞ Qn = − ρ f ⎜⎜ ⎟ − iων f ⎟⎟ ⎜ ν 4h ⎜⎝ ⎝ nπ ⎠ tanh kn( ) h ⎠⎜ ⎜ 1− ν kn( ) h ⎝ 2

(

)

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

[7.320]

For poorly viscous fluids like air or water, ν f is generally so small that the imaginary term within the first parentheses in the above formula can be safely neglected. Assuming it is the case, the generalized force [7.320] takes on the simple and remarkable following form:

(

Qn = −ω 2 M a Cν U n( ) − U n( 1

2)

)

[7.321]

M a stands for the added mass per unit plate length (Oz direction) which is the same as in absence of viscosity and given by: 2

⎛ L ⎞ L Ma = ρ f ⎜ x ⎟ x ⎝ 2nπ ⎠ h

[7.322]

Effects related to viscosity is accounted for by the coefficient:

(

⎛ tanh kn(ν ) h Cν = ⎜ 1 − ν ⎜ kn( ) h ⎝

) ⎞⎟ ⎟ ⎠

−1

[7.323]

Furthermore, with the aid of results [7.311], the nondimensional thickness kn( ) h can be expressed in terms of the Stokes number Sν relevant to the problem: ν

686


( k ( )h ) ν n

2

= ( kn h ) + i 2

ωh2 ωh2 S ν i = iSν ⇒ kn( ) h = (1 + i ) ν νf νf 2

[7.324]

The real part Rν of Cν is positive. It accounts for the added mass effect as modified by viscous transport. The imaginary part I ν of Cν is negative. It accounts to the dissipative effect of viscous shearing. The coefficients Rν and Iν are plotted in Figure 7.63 versus Sν = ω h 2 / ν f . As can be seen in the left-hand semilogarithmic plot, Rν is pratically equal to 1.2 if Sν is less than 10 and tends asymptotically to 1 as Sν tends to infinity. To plot Iν versus Sν , logarithmic scales in both axes are preferred since the range of variation is by far much larger than for Rν . As made conspicuous in the righ-hand side plot, I ν varies as 1/ Sν in the range Sν >1.

Figure 7.63. Real and imaginary parts of the viscous force coefficient Cν

The oscillator equation for the n-th mode in phase opposition is of the type:

(K

(s)

n

(

− iω 2 M n( )Iν − ω 2 M n( ) + M n( )Rν a

s

a

)) (U ( ) − U ( ) ) = 0 1 n

2

n

[7.325]

where K n( ) and M n( ) are the modal stiffness and mass coefficients of one plate in vacuum. Whence the equivalent viscous damping ratio: s

ς n( ) = ν

s

M n( )Rν Iν a

(

2 M n( ) + M n( )Rν s

a

)

In the range Sν >1, it simplifies practically into:

[7.326]


ς n( ) =

M n(

ν

(

a)

(s)

νf (a )

2 Mn + Mn

)

2ωn h 2

dQ T

[A1.17]

Keeping in the domain of reversible transitions, with the aid of [A1.16], relations [A1.3] and [A1.5] can be written as: TdS = dE + Pdυ

[A1.18]

TdS = dH − υ dP

[A1.19]

Now, if T and P are selected as the independent variables, the elemental change of entropy related to elemental changes in temperature and pressure is: ⎛ ∂S ⎞ ⎛ ∂S ⎞ dS = ⎜ ⎟ dT + ⎜ ⎟ dP ∂ T ⎝ ⎠P ⎝ ∂ P ⎠T

[A1.20]

As a definition, an isoentropic change is such that dS = 0. By virtue of [A1.16], an isoentropic process is also adiabatic. However, by virtue of [A1.17], the reverse is also true only if the adiabatic process is reversible. For instance, adiabatic free expansion of a gas is not isoentropic, while an ideal heat engine in which the working fluid would undergo adiabatic reversible cyclic process would be isoentropic. If the change is reversible, making use of [A1.19] and [A1.10], we obtain: ⎛ ∂S ⎞ CP ⎜ ⎟ = ⎝ ∂T ⎠ P T

[A1.21]

To transform the other partial derivative appearing in [A1.20] in terms of measurable quantities, a few non trivial manipulations are required, known as the Maxwell relations. A1.4

Maxwell relations

The starting point is to consider a closed cycle, i.e. a transformation such that the final state is the same as the initial one. Thus internal energy remains unchanged and using [A1.3], the following integral relation is verified:

∫ dE = ∫ dQ − ∫ Pdυ = 0

[A1.22]

The cycle being reversible, one find the expected result that if the system is assumed to return back to the initial state, the amount of heat delivered to it must be equal to the work performed by the system. Now, using [A1.16], relation [A1.22] implies:

712


∫ TdS = ∫ Pdυ

[A1.23]

which means that the closed paths followed by the system in the T, S and in the P, υ spaces are enclosing the same area, as sketched in Figure A1.1.

Figure A1.1. Closed cycles of equal areas in the spaces of variables T, S and P, υ

A mathematical consequence of [A1.23] is that the determinant of the Jacobian matrix related to the transformation of variables is unity: ⎡ ⎛ ∂S ⎞ ⎢⎜ ⎟ ∂ ( S , T ) ⎢ ⎝ ∂ P ⎠υ = [ J tr ] = ∂ ( P,υ ) ⎢⎛ ∂T ⎞ ⎢⎜ ⎟ ⎢⎣⎝ ∂ P ⎠υ

⎛ ∂S ⎞ ⎤ ⎜ ⎟ ⎥ ⎝ ∂υ ⎠ P ⎥ ; det [ J tr ] = 1 ⎛ ∂T ⎞ ⎥ ⎜ ⎟ ⎥ ⎝ ∂ υ ⎠ P ⎥⎦

[A1.24]

⎛ ∂S ⎞ To transform ⎜ ⎟ it is appropriate to consider the following Jacobian defined as: ⎝ ∂ P ⎠T

⎡ ⎛ ∂S ⎞ ⎢⎜ ⎟ ∂ ( S , T ) ⎢ ⎝ ∂ P ⎠T = [ J1 ] = ∂ ( P, T ) ⎢⎛ ∂T ⎞ ⎢⎜ ⎟ ⎣⎢⎝ ∂ P ⎠T

⎛ ∂S ⎞ ⎤ ⎜ ⎟ ⎥ ⎡ ⎛ ∂S ⎞ ⎝ ∂T ⎠ P ⎥ ⎢⎜ ⎟ = ⎢ ⎝ ∂ P ⎠T ⎥ ⎛ ∂T ⎞ ⎜ ⎟ ⎥ ⎢⎣ 0 ⎝ ∂ T ⎠T ⎦⎥

⎛ ∂S ⎞ ⎤ ⎛ ∂S ⎞ ⎜ ⎟ ⎥ ⎝ ∂T ⎠ P ⎥ ⇒ det [ J 1 ] = ⎜ ⎟ ⎝ ∂ P ⎠T ⎥ 1 ⎦

[A1.25] Using [A1.24] and noting that the reasoning which leads to det [ J tr ] = 1 is the same whether the transformation S , T → P,υ or the inverse P,υ → S , T is considered, relation [A1.25] is suitably transformed into: ⎡ ∂ ( S , T ) ∂ ( P,υ ) ⎤ ⎡ ∂ ( P,υ ) ⎤ ⎛ ∂S ⎞ ⎥ = det ⎢ ⎥ ⎜ ⎟ = det ⎢ , , ∂ ∂ ∂ P P T S T ) ( )⎦ ⎝ ⎠T ⎣ ( ⎣ ∂ ( P, T ) ⎦

[A1.26]

Appendices

713

which gives the desired result: ⎡ ⎛ ∂P ⎞ ⎢⎜ ⎟ ⎝ ∂ P ⎠T det ⎢ ⎢⎛ ∂υ ⎞ ⎢⎜ ⎟ ⎢⎣⎝ ∂ P ⎠T

A1.5

⎛ ∂P ⎞ ⎤ ⎡ 1 ⎜ ⎟ ⎥ ⎝ ∂T ⎠ P ⎥ ⎢ = det ⎢⎛ ∂υ ⎞ ⎛ ∂υ ⎞ ⎥ ⎢⎣⎝⎜ ∂ P ⎠⎟T ⎜ ⎟ ⎥ ⎝ ∂ T ⎠ P ⎥⎦

⎤ ⎛ ∂S ⎞ ⎥ ⎛ ∂υ ⎞ ⎛ ∂υ ⎞ ⎥ = ⎜ ⎟ =⎜ ⎟ ⎜ ⎟ ⎝ ∂ T ⎠ P ⎝ ∂ P ⎠T ⎝ ∂ T ⎠ P ⎥⎦ 0

[A1.27]

Thermodynamic relations particularized to perfect gases

For low enough pressures, real gases are in general well described by the socalled perfect gas model, which assimilates molecules with material points without mutual interactions other than elastic shocks. Each molecule then travels along a straight line until impacting another molecule or a solid wall, where it rebounds elastically. Application of such kinetic theory enables one to find Boyle's law, written under the following equivalent forms: ρ=

PM ⇔ PV = nR T RT

[A1.28]

where ρ is the gas density, P the pressure and T the absolute temperature (in ºK), M is the molecular mass, V is the volume occupied by the gas, n is the number of moles in V and R = 8.314 Joule/mole °K designates the universal gas constant. We recall that a mole contains the so-called Avogadro number of molecules, N A = 6.02 1023 . The perfect gas law for a unit mass is written as: Pυ =

P RT = ρ M

Therefore, in a perfect gas internal energy and enthalpy are functions of the temperature alone. With the aid of [A1.28], the coefficient of thermal expansion at constant pressure is easily found to be: 1 β= [A1.29] T The relation [A1.9] becomes CP − CV =

R M

[A1.30]

The rate of increase of pressure with temperature at constant volume becomes: α=

P = ρ 0R T

[A1.31]

Considering now an adiabatic process, with the aid of [A1.3] to [A1.5] the following relationship can be specified between the pressure and the volume changes:

714


dQ =

⎛ ∂H ∂Q ∂Q dυ + dP = 0 ⇒ ⎜ ∂υ P ∂P υ ⎝ ∂T

P

⎛ ∂E ∂T ⎞ ∂T ⎞ ⎟ dP = 0 ⎟ dυ + ⎜ ∂υ P ⎠ ⎝ ∂T υ ∂υ υ ⎠

[A1.32]

With the aid of [A1.6], [A1.7] and [A1.28] relation [A1.32] is further transformed into: CP

υd P Pdυ d P⎫ ⎧ dυ + CV = 0 ⇔ T ⎨C P + CV ⎬=0 R R υ P ⎭ ⎩

[A1.33]

Relation [A1.33] can also be read as: γ

⎛P⎞ dP dV ⎛V ⎞ = −γ ⇒ Log ⎜ ⎟ = Log ⎜ 0 ⎟ ⇒ PV γ = P0V0γ P P V ⎝V ⎠ ⎝ 0⎠

[A1.34]

where V is the volume occupied by the gas, not necessarily a unit mass. Let us consider now a non adiabatic transformation. From [A1.32] and [A1.33], it is immediately derived that: dQ dV dP = CP + CV T P V

[A1.35]

For an arbitrary volume of perfect gas the law [A1.28] has simply to be multiplied by the number of moles contained in the volume. It is thus immediate to check that the following relation holds for any quantity of perfect gas: dT dV dP = + T V P

[A1.36]

Using [A1.36], [A1.35] is transformed into: dQ dV dT Td V = ( CP − CV ) + CV ⇒ dQ = ( CP − CV ) + CV d T T T V V

[A1.37]

which is finally written as: dQ = ( CP − CV )

Pd V + CV d T nR

[A1.38]

Where n designates here the number of moles in the amount of gas considered. Two particular cases are of special importance. At first, for an isothermal process, with the aid of [A1.30], [A1.38] is found to reduce to the expected result concerning the amount of heat produced by an isothermal change of volume: dQ =

Pd V nM

[A1.39]

which fully agrees with [A1.3], where dE is set to zero and the unit mass replaced by the actual mass nM of the amount of fluid considered.

Appendices

715

The second case, concerns the change in temperature related to an adiabatic compression, or expansion. From [A1.38], the change in temperature is first written as: Pd V [A1.40] d T = − (γ − 1) nR Using the adiabatic law [A1.34], [A1.40] is transformed into: γ − 1 V dP dT = nR γ

[A1.41]

relation which is rewritten in its final form by using the perfect gas law [A1.28]: dT γ − 1 dP [A1.42] = T γ P To relate pressure to volume for processes intermediate between the isothermal and adiabatic cases, it is convenient to use the so-called polytropic law written as: PV

γp

= constant

where the polytropic index γ p

[A1.43] can be any real number. As particular cases of

special importance, the process is isobaric if p = 0, isothermal if γ p = 1 , isovolumetric, or isochoric, if γ p tends to infinity and finally adiabatic if γ p = γ . A1.6

Heat transfer and energy losses

Let E denote the total energy of a fluid which is contained in a fixed volume (Vf ) . In any thermodynamical transformation, reversible or not, the rate of change of the total fluid energy must be balanced by three distinct terms, namely: 1 – The energy flow through the boundary (S f ) of the control volume: ⌠ J E = ⎮⎮

⌡(S f )

jE .n dS

[A1.44]

where the unit vector n ( r ) normal to (S f ) is conventionally pointing outward the

control volume. The energy flux vector is defined as: jE = ρ eV [A1.45] where V is the Eulerian fluid velocity and e stands for the total energy of the fluid per unit mass. 2 – The heat flux through (S f ) : ⌠

J H = ⎮⎮

⌡(S f

)

jH .n dS

[A1.46]

716


where jH is the heat flux vector as introduced in Chapter 4.

3 – The work on the fluid due to the surface forces (here the pressure on (S f ) ): ⌠

WP = ⎮⎮

⌡(S f

)

PV.n dS

[A1.47]

The energy balance on the control volume is thus written as: ⌠ ⌠ ⌠ ∂ ⌠ ⎮ ρ edV + ⎮⎮ ρ eV.n dS + ⎮⎮ PV.n dS + ⎮⎮ jH .n dS = 0 ⎮ ⌡(S f ) ∂ t ⌡(Vf ) ⌡(S f ) ⌡(S f )

[A1.48]

Using the divergence theorem and the fact that (Vf ) can be chosen at will, [A1.48] can be replaced by the local equation: ∂ ( ρe ) + div ⎡⎣( ρ e + P )V + jH ⎤⎦ = 0 ∂t

[A1.49]

Equation [A1.49] is further transformed by expanding the first two terms: ∂ ( ρe ) ∂ρ ∂e + div ⎡⎣ ρ eV ⎤⎦ = e + ρ + ρ e div V + eV .grad ρ ∂t ∂t ∂t

[A1.50]

Using the continuity equation [1.13], the first term of [A1.50] can be written as: ∂ρ [A1.51] = − ρ e div V − eV .grad ρ e ∂t by substituting [A1.51] into [A1.50], we arrive at: ∂ ( ρe ) ∂e De + div ⎡⎣ ρ eV ⎤⎦ = ρ + ρV .grad e = ρ Dt ∂t ∂t

At this step, the energy equation [A1.49] can be written as: De ρ + div ⎣⎡ PV + jH ⎦⎤ = 0 Dt

[A1.52]

[A1.53]

Equation [A1.53] is particularized to the case where the fluid energy per unit mass can be written as the sum of an internal energy term, and a kinetic energy term: e=E+

V2 2

[A1.54]

whence, ρ

∂V De DE =ρ + ρV . + ρV .grad (V 2 / 2 ) Dt Dt ∂t

[A1.55]

Appendices

717

On the other hand, with the aid of the momentum equation [1.43], particularized to the case of an inviscid flow for convenience, the first term due to the kinetic energy is written as: ∂V ρV . = − ρV . ⎛⎜ V .gradV ⎞⎟ − V .gradP [A1.56] ∂t ⎝ ⎠ Substituting [A1.56] into [A1.55] yields: De DE ρ =ρ + −V .gradP Dt Dt With the aid of [A1.57], the energy equation [A1.53] reads as: DE ρ + P div V + div jH = 0 Dt

[A1.57]

[A1.58]

Assuming conduction is the relevant heat transfer mechanism, the heat vector flux is governed by the Fourier law: jH = −κ H grad T [A1.59] where κ H is the thermal conduction coefficient and T the absolute temperature, as explained in Chapter 4, section 4.4. Substituting [A1.59] into [A1.58] we finally obtain: DE ρ + P div V − κ H ΔT = 0 [A1.60] Dt Due to thermal conduction, a thermodynamical process cannot be reversible. Therefore change in entropy is expected to be larger than the amount which is derived by using the transformation law [A1.18], rewritten here as: dS dE P T = + divV [A1.61] dt dt ρ where use is made of the physical interpretation of divX f in terms of volume variation: d d ⎛ dυ ⎞ d divV = divX f = ⎜ ⎟ = ( ρ dυ ) dt dt ⎝ υ ⎠ dt

(

)

[A1.62]

Substituting the linearized version of [A1.60] into [A1.61], one obtains: ρ

dS κ H = Δ [T ] dt T

[A1.63]

Appendix A2

Mechanical properties of common materials In this appendix we present a data set corresponding to the mechanical properties of several common gases and liquids, borrowed in particular from the extensive compilation [BLE90], as well as some physical properties of common solid materials [FAH01]. A2.1

Phase diagram

Figure A2.1. Phase diagram of a pure substance

Figure A2.1 displays a sketch of the pressure and temperature domains where the solid, liquid and vapour phases of a pure substance exist. The triple point corresponds to the pair of pressure and temperature values such that the three phases coexist in thermodynamical equilibrium. The fusion curve is bent to the right or to

Appendices

719

the left, depending on the substance contracts or expands when it freezes. Along the boundaries of sublimation and vaporization, the vapour pressure is equal to the atmospheric pressure. The vaporization curve ends at the critical point. At pressures higher than the critical pressure, one cannot distinguish between a vapour and a liquid phase. On the other hand, it is impossible to liquefy vapour if the temperature is higher than the critical temperature. A2.2

Gas properties

The physical properties of several common gases are presented in Table A2.1. Within a large domain of pressure and temperature, the viscosity of most real gases is obtained from F P / Pc ; T / Tc diagram. The coefficient of dynamic viscosity is independent of pressure. It changes with temperature according to the law of Chapman and Enskog:

b

μ = 2 6 .6 9 1 0 − 7

a f

Θ T = 1.1 4 7

g

MT σ 2Θ T

FTI GH Tε JK

a f

− 0 .1 4 5

+ 1.1 4 7

FT GH Tε

+ 0 .5

I −2 JK

[A2.1]

where σ is the collision diameter, expressed in Aº, and Tε is the effective temperature of the force potential, in °K (see Table A2.2). For instance, in the case of air at normal atmospheric conditions, T = 18 0 C , P = 1.01105 Pa , we have μ = 17.6 10−6 Nms −1 = 0.177 milliPoise and ν = 15 10−6 m 2 s −1 = 0.15 Stoke . Table A2.1. Physical properties of some gases

M

Substance

γ

µc x106

Tc (°K)

(Ns/m )

Pc (atm)

2

ρ0

–3

c0

(kgm ) (ms–1)

Air

*

29.0

1.40

19.3

60.6

308

1.21

343

Ammonia

NH3

17.0

1.31

32.7

111.0

406

–

440

Argon

Ar

39.9

1.66

26.4

48.1

151

–

308

Nitrogen

N2

28.0

1.4

18.0

33.5

126.2

–

354

Butane

C4H10

58.1

1.09

25.0

37.5

425

–

–

Carbon dioxide

CO2

44.0

1.29

34.3

72.8

304.2

1.98

280

Carb. monoxide

CO

28.0

1.4

19.0

34.5

132.9

–

–

Helium

He

4.0

1.67

25.4

2.24

5.19

0.18

973

Hydrogen

H2

2.02

1.4

3.47

12.8

33.2

0.09

1270

Methane

CH4

16.04

1.32

15.9

45.4

191

–

466

720


M

Substance

γ

µc x106

Tc (°K)

(Ns/m )

Pc (atm)

2

ρ0

c0

–3

(kgm ) (ms–1)

Neon

Ne

20.18

1.67

16.3

27.2

44.4

–

461

Octane

C8H18

114.2

1.04

24.1

24.5

563

–

–

Oxygen

O2

32.0

1.40

25.0

49.8

155

1.43

332

Propane

C3H8

44.1

1.12

23.3

41.9

370

–

–

Steam water

H2O

18.1

1.33

54.1

218.

647

0.6

405

Xenon

Xe

131.3

1.66

53.7

57.6

290

–

–

Table A2.2 shows the Chapman and Enskog model parameters, for several gases. Table A2.2. Parameters of the viscosity model by Chapman and Enskog Substance

σ (A°)

Tε (°Κ)

Substance

σ (A°)

Tε (°Κ)

Air

3.711

78.6

H

2.827

59.7

NH3

2.900

558.3

CH4

3.758

148.6

Ar

3.542

93.3

Ne

2.820

32.8

N2

3.798

71.4

C8H10

–

–

C4H10

4.687

531.4

O2

3.467

106.7

CO2

3.941

195.2

C3H8

5.118

327.1

CO

3.690

91.7

H2O

2.641

809.1

He

2.551

10.2

Xe

4.047

231.0

A2.3

Liquid properties

There are no precise theoretical formulations for describing the properties of liquids. Within a good approximation c0 , ρ 0 , μ 0 only depend on temperature, at least below the critical pressure. The viscosity of many liquids change with temperature according to the empirical law of Van Velsen: μ 0 = 10 −310

b

A 1/ T −1/ Tc

g Nsm −2

[A2.2]

Tables A2.3 and A2.4 display relevant physical parameters, respectively for sea water and pure water, as a function of temperature.

Appendices

721

Table A2.3. Physical properties of sea water (salinity 3.5%) T (°C)

ρ 0 (kgm–3)

c0 (ms–1)

ν 0 (m 2s-1 )

0

2810

1449

1.83 10-6

5

2768

1471

1.56

10

2696

1490

1.35

15

2597

1507

1.19

20

2475

1521

1.05

25

2334

1534

0.946

30

–

1546

0.853

Table A2.4. Physical properties of pure water T (°C)

ρ 0 (kgm–3)

c0 (ms–1)

ν 0 (m 2s-1 ) -6

(

σ f Nm -1

0

999

1403

1.79 10

5

1000

1427

1.52

0.0749

10

999

1447

1.31

0.0742

15

999

1465

1.14

0.0735

20

998

1481

1.04

0.0727

25

997

1495

0.893

0.0720

30

995

1507

0.809

0.0712

40

992

1526

0.658

0.0696

50

988

1541

0.553

0.0679

60

983

1552

0.475

0.0662

70

977

1555

0.413

0.0644

80

971

1555

0.365

0.0626

90

965

1553

0.326

0.0608

100

958

1543

0.294

0.0589

)

0.0756

The physical properties of other liquids are presented in Tables A2.5 and A2.6.

722

Fluid-structure interaction Table A2.5. Physical properties of alcohols and oils T ref °K

ρ 0 (kgm–3)

c0 (ms–1)

ν 0 (m 2s-1 )

A

T0

- ethyl

20

789

1159

15.2

687

301

- butyl

20

810

1263

36.4

985

341

- methyl

20

791

1120

7.55

555

261

- isopropyl

20

786

1170

31.7

1140

323

- castor oil

20

950

1540

10300

–

–

- crude

20

850

1326

70

–

–

- SAE 30

20

920

1290

730

–

–

Substance Alcohols:

Oils:

Table A2.6. Physical properties of various liquids T ref °K

ρ 0 (kgm–3)

c0 (ms–1)

ν 0 (m 2s-1 )

A

T0

20

932

1152

1020 x 10-7

–

–

–188

849.4

–

2.02

–

–

Acetone

20

790

1189

4.13

367

210

Nitrogen

–195

804

894

1.89

90

46

Benzene

16

885

1324

7.37

546

265

Dioxide of carbon

20

777

856

0.91

578

185

Bisulfite of carbon

0

1293

1157

2.81

274

200

Gasoline

20

670

1395

4.6

–

–

Ethylic ether

20

713

1006

3.27

353

191

Glycerine

20

1231

1895

11800

3337

406

Helium

–269

129

181

0.254

–

–

Hydrogen

–253

71

1100

1.52

13.8

5.39

Kerosene

20

804

1460

23.0

–

–

Mercury

20

13600

1451

1.2

–

–

Methane

–163

425

1380

2.49

114

57.6

Octane

20

971

–

8.97

873

353

Oxygen

–183

1149

1159

2.32

85.7

51.5

Substance Acetate of Vinyl Air

Appendices

723

T ref °K

ρ 0 (kgm–3)

c0 (ms–1)

ν 0 (m 2s-1 )

A

T0

Sodium

225

897

2461

4.72

–

–

Turpentine

20

870

1330

17.1

–

–

Substance

A2.4

Solid properties

Finally, Tables A2.7 and A2.8 display relevant properties of several solids and common metals. Table A2.7. Physical properties of several solids Substance

E (Nm–2)

ρ 0 (kgm–3)

Poisson ratio

c0 (ms–1)

Glass

6.0 1010

2400

0.24

5000

1300

–

1700

2300

–

3360

950

0.5

70

Light concrete Dense concrete Soft rubber

3.8 10 2.6 10

9

10

5.0 10

6 9

Hard rubber

2.3 10

1100

0.4

1450

Brick

1.6 1010

2000

–

2800

Dry sand

3.0 107

1500

–

140

1200

–

2420

650

–

2660

1200

0.4

2160

600

–

3000

Plaster Chipboard Perspex Plywood

7.0 10 4.6 10 5.6 10 5.4 10

9 9 9 9

Cork

–

120~240

–

430

Asbestos cement

2.8 1010

2000

–

3700

Table A2.8. Physical properties of common metals Substance

E (Nm–2)

ρ 0 (kgm–3)

Poisson ratio

c0 (ms–1)

Aluminium

7.1 1010

2700

0.33

5130

8500

0.36

3430

8900

0.35

3750

7800

0.28

5060

Brass Copper Steel

1.0 10 1.3 10 2.0 10

11 11 11

Appendix A3

The Green identity A brief proof of the Green identity will be presented here. Let F ( x, y, z ) be a vector field defined within a volume V delimited by the surface S . Then Gauss’s theorem states that: ⌠ ⎮ ⎮ ⌡V

⌠ div F dV = ⎮⎮ F . n dS

[A3.1]

⌡S

showing that a closed volume integral can be reduced to a closed surface integral. This relation holds for a space of any dimensionality. Hence, in the one-dimensional version of the Gauss theorem, the vector field has a single component and the “volume” where Fx ( x) lies is simply a line, running from the boundaries x = a to x = b . Then [A3.1] implies the easily recognized formula: b ⌠ ⎮ ∂Fx ⎮ ⎮ ⎮ ∂x ⌡a

dV = Fx (b) − Fx ( a )

[A3.2]

We now assume that X 1 ( x, y, z ) and X 2 ( x, y, y ) are two scalar fields defined within the same domain, and recall the vector identities: div X 1 grad X 2 = grad X 1 . grad X 2 + X 1 ΔX 2 [A3.3] div X 2 grad X 1 = grad X 2 . grad X 1 + X 2 ΔX 1

( (

) )

of which we take the difference and integrate within volume V : ⌠ ⎮ ⎮ ⎮ ⌡V

⌠ ⎡div X 1 grad X 2 − div X 2 grad X 1 ⎤ dV = ⎮ [ X 1 ΔX 2 − X 2 ΔX 1 ] dV ⎮ ⎣ ⎦ ⌡V

(

)

(

)

[A3.4]

Appendices

725

If we now apply the Gauss theorem to [A3.4], with F = X 1 grad X 2 − X 2 grad X 1 , the left hand side of this equation becomes: ⌠ ⎮ ⎮ ⎮ ⌡V

⎡div X 1 grad X 2 − div X 2 grad X 1 ⎤ dV = ⎣ ⎦

(

)

=

(

⌠ ⎮ ⎮ ⎮ ⌡S

(

)

[A3.5]

X 1 grad X 2 − X 2 grad X 1 . n dS

)

and, from [A3.4] and [A3.5], we obtain the Green identity: ⌠ ⎮ ⎮ ⌡V

[ X 1 ΔX 2 − X 2 ΔX 1 ] dV

⌠

= ⎮⎮

⎮ ⌡S

(X

1

grad X 2 − X 2 grad X 1 . n d S

)

[A3.6]

Appendix A4

Bessel functions This appendix supplies basic information pertaining to the various types of Bessel functions, as well as their properties. More information on this topic may be found in many books on applied mathematics and special functions, for instance [BOW 58], [ANG 61], [ABR 84] or [WAT 95]. A4.1

Definition

The Bessel functions of the first and second kind, of order ν , are the particular solutions of the so-called Bessel differential equation:

F GH

I JK

ν2 d 2 y 1 dy + + 1− 2 y = 0 2 x dx dx x

[A4.1]

where ν is any given number. The general solution of [A4.1] is: y ( x) = AJν ( x ) + BYν ( x )

[A4.2]

where Jν ( x ) designates the Bessel function of the first kind and Yν ( x ) the Bessel function of the second kind, of order ν . A4.2

Bessel functions of the first kind

The Bessel functions of the first kind, Jν ( x ) , are written in series form as: ∞

Jν ( x ) = xν ∑ aλ x λ λ =0

with Re (ν ) > 0

[A4.3]

One finds: 2λ ( −1) ⎛ x⎞ ∑ ⎜ ⎟ λ = 0 λ !Γ (1 + λ + ν ) ⎝ 2 ⎠ λ −ν 2λ ( −1) ⎛x⎞ ∞ ⎛x⎞ J −ν ( x ) = ⎜ ⎟ ∑ ⎜ ⎟ ⎝ 2 ⎠ λ = 0 λ !Γ (1 + λ −ν ) ⎝ 2 ⎠ ν

⎛x⎞ Jν ( x ) = ⎜ ⎟ ⎝2⎠

∞

λ

[A4.4]

Appendices

727

af

where use is made of the factorial Gamma function Γ x , which may be defined as:

F GH

af

p! p x Γ x = lim p→∞ x ( x + 1)( x + 2)...( x + p)

I JK

[A4.5]

af

Figure A4.1 shows several plots of the J n x functions, for x positive and n integer. All these functions become nil at x=0, except function J 0 x . They oscillate with a decreasing amplitude as x increases. The zeros of J n x and J n+1 x are interlaced. The functions of even order are even and odd functions are odd. If the order ν is nonintegral it can be shown that functions Jν ( x ) and J −ν ( x ) are

af af

af

linearly independent. They can thus be used as basis functions to construct the general solution of the Bessel equation [A4.1]. However, if ν is integer, we have the relation:

af a f af

J − n x = −1 n J n x

[A4.6]

Figure A4.1. Bessel functions of the first kind with integer order ν

A4.3

Bessel functions of the second kind

The Bessel functions of the second kind, Yν ( x ) , are defined by the relation: Yν ( x ) =

cos (πν ) Jν ( x ) − J −ν ( x ) sin (πν )

[A4.7]

728


If ν is not integral, this function is a particular solution of equation [A4.1], which arises as a linear form of the base functions Jν ( x ) and J −ν ( x ) . If ν is

af

integer, it can be shown that Yn x – which is given by relation [A4.7] in indeterminate form – is linearly independent of J n x .

af

Figure A4.2. Bessel functions of the second kind with integer order ν

af

Figure A4.2 shows several plots of the Yn x functions, for x positive and n integer. All these functions tend to −∞ when x tends to zero, and they oscillate with decreasing amplitude as x increases. The zeros of Yn x and Yn+1 x are also interlaced. Again, functions of even order are even and odd functions are odd. A4.4

af

af

Recurrence relations

2ν Jν ( x ) = Jν −1 ( x ) + Jν +1 ( x ) x Jν′ ( x ) = Jν −1 ( x ) − Jν +1 ( x ) J 0′ ( x ) = − J1 ( x )

[A4.8]

af

af

The zeros of function J1 x correspond to the extrema of J 0 x . Functions Yν ( x ) comply with the same recurrence relations as Jν ( x ) . Obviously, the same

applies to all linear combinations AJν ( x ) + BYν ( x ) .

Appendices

A4.5

729

Remarkable integrals

The previous recurrence relations provide the following results: x

⌠ 2 ⎮ ⎮ ⌡ x1 x

⌠ 2 ⎮ ⎮ ⌡ x1 x

⌠ ⎮ ⎮ ⌡0

x2

xν Jν −1 ( x ) dx = ⎡⎣ xν Jν ( x ) ⎤⎦ x1

[A4.9]

x2

x −ν Jν +1 ( x ) dx = − ⎡⎣ x −ν Jν ( x ) ⎤⎦ x1

[A4.10]

Jν ( x ) dx = 2 { Jν +1 ( x ) + Jν + 3 ( x ) + ...}

[A4.11]

A4.6

Lommel integrals

For k ≠ , we have: x

⌠ ⎮ ⎮ ⌡0

x

⌠ ⎮ ⎮ ⌡0

x

⌠ ⎮ ⎮ ⎮ ⌡0

xJν ( kx ) Jν ( x ) xdx =

z {kJν +1 ( kx ) Jν ( x ) − Jν +1 ( x ) Jν ( kx )} k − 2

[A4.12]

xJν ( kx ) Jν ( x ) xdx =

x {Jν −1 ( x ) Jν ( kx ) − kJν −1 ( kx ) Jν ( x )} k 2 − 2

[A4.13]

( J ( kx ) )

2

ν

A4.7

xdx =

x2 2

2

⎧⎪ ⎛ 2 2⎫ ν2 ⎞ ⎪ ⎨( Jν′ ( kx ) ) + ⎜1 − 2 2 ⎟ ( Jν ( kx ) ) ⎬ k x ⎝ ⎠ ⎩⎪ ⎭⎪

[A4.14]

Hankel functions

Hankel functions are to Bessel functions what exponential functions are to trigonometric functions. They are defined through the relations:

af af af H a f a x f = J a x f − iY a x f Hνa1f x = Jν x + iYν x ν

2

ν

[A4.15]

ν

The Hankel functions are often also referred in the literature as Bessel functions of the third kind. A4.8

Asymptotic forms for large values of the argument

If x is large enough and if −π / 2 < arg( x) < π / 2 , we have:

730


Jν ( x ) ≅

2π cos ϕ ; Yν ( x ) ≅ x 2π iϕ e x

Hν( ) = 1

; Hν( ) = 2

with ϕ = x − ( 2ν + 1)

A4.9

2π sin ϕ x

2π −iϕ e x

[A4.16]

π 4

Modified Bessel functions of the first and second kinds

Replacing x by ix in the Bessel differential equation [A4.1], we find that Jν ( ix ) is the solution of the following equation: d 2 y 1 dy ⎛ ν 2 + − ⎜1 + dx 2 x dx ⎝ x 2

⎞ ⎟y=0 ⎠

[A4.17]

One then defines the modified Bessel function of the first kind through the relation: Iν ( x ) = i −ν Jν ( ix )

[A4.18]

and, if ν = n is integer, we have: I −n ( x ) = I n ( x )

[A4.19]

We now examine the function: Kν ( x ) =

π I −ν ( x ) − Iν ( x ) 2 sin (πν )

[A4.20]

which is also a solution of equation [A4.17]. It can be shown that, if ν tends to an integer value n, then Kν ( x ) tends to K n ( x ) which is linearly independent of I n ( x ) . We call Kν ( x ) a modified Bessel function of the second kind. It can be

expressed simply in terms of the Hankel functions: Kν ( x ) = iν +1

π (1) Hν ( ix ) 2

[A4.21]

If x is large enough, we have the asymptotic relations: Iν ( x ) ≈

ex 2π x

; Kν ( x ) ≈

π −x e 2x

[A4.22]

Appendices

Figure A4.3. Modified Bessel functions of the first kind with integer order ν

Figure A4.4. Modified Bessel functions of the second kind with integer order ν

731

Appendix A5

Spherical functions The same way as the Bessel functions presented in Appendix A4 are connected with solutions pertaining to cylindrical geometries, the functions whose properties are described in the following naturally arise when solving field problems described in spherical coordinate systems. For additional information on this topic see for instance [ABR 84], [ANG 61]. A5.1

Legendre functions and polynomials

The Legendre functions Pnm ( x) , of degree n and order m, are solutions of the differential equation: (1 − x 2 )

d2y dy ⎡ m2 ⎤ − 2 x + ⎢ n ( n + 1) − ⎥y=0 2 dx dx ⎣ 1 − x2 ⎦

[A5.1]

where the maximum order for a given degree is m ≤ n . In general, our interest is restricted to real arguments in the range x ≤ 1 and, when dealing with spherical waves, x = cos ϕ . As an important case, the so-called Legendre polynomials Pn ( x) ≡ Pn0 ( x) are zero-order Legendre functions, which are solutions of: (1 − x 2 )

d2y dy − 2 x + n ( n + 1) y = 0 2 dx dx

[A5.2]

Solution of [A5.2] can be thought in the form of a power series: ∞

y ( x) = ∑ ci x r

[A5.3]

r =0

leading to: (1 − x 2 )(2c2 + 6c3 x + 12c4 x 2 + ) − 2 x(c1 + 2c2 x + 3c3 x 2 + ) + + n ( n + 1) (c0 + c1 x + c2 x 2 + ) = 0

[A5.4]

Appendices

733

and, collecting terms: ⎣⎡ 2c2 + n ( n + 1) c0 ⎦⎤ + ⎣⎡6c3 − 2c1 + n ( n + 1) c1 ⎦⎤ x + + ⎡⎣12c4 − 6c2 + n ( n + 1) c2 ⎤⎦ x 2 + = 0

[A5.5]

whence: n ( n + 1)

1 c0 = − n ( n + 1) c0 2 2! 2 − n ( n + 1) 1 c3 = c1 = ⎡⎣ 2 − n ( n + 1) ⎤⎦ c1 6 3! 6 − n ( n + 1) 1 c4 = c2 = − ⎡⎣ 6 − n ( n + 1) ⎤⎦ c0 12 4! c2 = −

[A5.6]

and so on. Therefore, from [A5.3] and [A5.6] the solution of the Legendre equation reads: 1 ⎡ 1 ⎤ y ( x; n) = ⎢1 − n ( n + 1) x 2 − ⎡⎣ 6 − n ( n + 1) ⎤⎦ x 4 − ⎥ c0 + 2! 4! ⎣ ⎦ [A5.7] 1 1 ⎡ ⎤ 3 5 + ⎢ x + ⎡⎣ 2 − n ( n + 1) ⎤⎦ x + ⎡⎣ 2 − n ( n + 1) ⎤⎦ ⎡⎣12 − n ( n + 1) ⎤⎦ x + ⎥ c1 3! 5! ⎣ ⎦

where c0 and c1 are arbitrary constants. These solutions may be shown to converge

within the interval [ −1 1] . If c1 = 0 , only the first series remains while if c0 = 0 only the second series remains. Furthermore, in general: cr + 2 =

r ( r + 1) − n ( n + 1) (r + 2)(r + 1)

cr

[A5.8]

which implies that only a finite number of terms of the series occur. If n is even, c2 r + 2 = c2 r + 4 = = 0 , while if n is odd, c2 r + 3 = c2 r + 5 = = 0 . We thus obtain the so-called Lagrange polynomials Pn ( x) ≡ y ( x; n) , defined as: N

Pn ( x) = ∑ (−1)r r =0

(2n − 2r )! xn−2r 2n r !(n − r )!(n − 2r )!

[A5.9]

with N = n / 2 if n is even or N = ( n − 1) / 2 if n is odd. In alternative, Legendre polynomials may be generated using Rodrigues’s formula: 1 dn 2 ( x − 1) n [A5.10] 2n n ! dx n while the higher order associated Legendre functions can be obtained from: Pn ( x) =

734


dm Pn ( x) [A5.11] dx m In Figure A5.1 are shown the six lowest degree Legendre polynomials:

Pnm ( x) = (−1) m (1 − x 2 ) m / 2

P0 ( x) = 1 1 P2 ( x) = ( 3x 2 − 1) 2 1 P4 ( x) = ( 35 x 4 − 30 x 2 + 3) 8

P1 ( x) = x 1 P3 ( x) = ( 5 x3 − 3x ) 2 1 P5 ( x) = ( 63x5 − 70 x3 + 15 x ) 8

[A5.12]

Notice that he even-numbered polynomials are symmetrical while the oddnumbered are anti-symmetrical. On the other hand, all Pn ( x) amplitudes become unity at x = 1 .

Figure A5.1. Legendre polynomials of even and odd degrees

Appendices

A5.2

735

Recurrence and orthogonality relations for Legendre polynomials

Legendre polynomials of a given degree may be computed from the preceding degrees using the following recurrence: Pn +1 ( x) =

2n + 1 n xPn ( x) − Pn −1 ( x) n +1 n +1

[A5.13]

Legendre polynomials form a complete set of orthogonal functions: n≠s ⎧ 0 ⎪ ∫ Pn ( x) Ps ( x) dx = ⎨ 2 n = s −1 ⎪⎩ 2n + 1 1

[A5.14]

and therefore may be used in series expansions over the interval [ −1 1] . A5.3

Spherical Bessel functions

As shown in Chapter 5, the radial equation in spherical problems leads to the socalled spherical Bessel equation, of the form: d 2 w 2 dw ⎡ 2 n(n + 1) ⎤ + + k − w=0 dx 2 x dx ⎢⎣ x 2 ⎥⎦

[A5.15]

which, after the transformation w( x) = x −1/ 2 y ( x) leads to a Bessel equation of halfinteger order ν = n + 1/ 2 : d 2 y 1 dy ⎡ 2 (n + 1/ 2)2 ⎤ + + ⎢k − ⎥y=0 dx 2 x dx ⎣ x2 ⎦

[A5.16]

The solution of [A5.16] can be written in terms of the ordinary Bessel functions of the first and second kind (the later also called Neumann functions): ⎧ ⎪ ⎪ y ( x; n, k ) = ⎨ ⎪ ⎪⎩

1 kx 1 kx

J n +1/ 2 (kx)

[A5.17] Yn +1/ 2 (kx)

which for half-integer orders can be expressed in terms of the so-called spherical Bessel functions: J n +1/ 2 (kx) = kx jn (kx ) ; Yn +1/ 2 (kx ) = kx yn (kx )

[A5.18]

hence the solution of the spherical Bessel equation may be stated in terms of the spherical functions:

736


⎧ j (kx) y ( x; n, k ) = ⎨ n ⎩ yn (kx)

[A5.19]

These can be expressed in series form: (r + n)! x2r r !(2r + 2n + 1)!

∞

jn ( x) = (2 x) n ∑ (−1)r r =0

yn ( x ) =

(−1)n +1 ∞ (r − n)! x2r (−1)r n ∑ x(2 x) r = 0 r !(2r − 2n)!

or using common trigonometric functions, we have: 1 j0 ( x) = sin x x 1 1 j1 ( x) = 2 sin x − cos x x x 3 1 3 ⎛ ⎞ j2 ( x) = ⎜ 3 − ⎟ sin x − 2 cos x x⎠ x ⎝x

[A5.20] [A5.21]

[A5.22]

and so on, as well as: 1 y0 ( x) = − cos x x 1 1 y1 ( x) = − 2 cos x − sin x [A5.23] x x 3 ⎛ 3 1⎞ y2 ( x) = − ⎜ 3 − ⎟ cos x − 2 sin x x⎠ x ⎝x A few low order spherical Bessel functions are shown in Figure A5.2. Notice that the spherical Bessel functions of the second kind yn ( x) are infinite at the origin, meaning they cannot be used to construct wave solutions when the origin is included, such as in the case of the acoustical field inside a sphere. A5.4

Recurrence relations for spherical Bessel functions

A first relation generates the complete family of spherical Bessel functions by successive derivatives of the first one: n

⎛1 d ⎞ jn ( x ) = ( − x ) n ⎜ ⎟ j0 ( x) ⎝ x dx ⎠

[A5.24]

while a second recurrence relation is: jn +1 ( x) =

2n + 1 jn ( x) − jn −1 ( x) x

[A5.25]

Appendices

737

and the first derivative of spherical Bessel functions are given by: jn′ ( x) =

1 [ n jn −1 ( x) − (n + 1) jn +1 ( x)] 2n + 1

[A5.26]

The same relations apply to the spherical functions yn ( x) .

Figure A5.2. Spherical Bessel functions of the first and second kinds

A5.5

Spherical Hankel functions

The same way as shown in Appendix A4 for ordinary Bessel functions, one can construct complex spherical Hankel functions from the spherical Bessel functions of the first and second kinds, thus obtaining:

738


hn(1) ( x) = jn ( x) + i yn ( x)

[A5.26]

hn(2) ( x) = jn ( x) − i yn ( x)

The following are useful relations for computing the spherical Hankel functions: hn(1) ( x) =

i − n ix n (n + s )! ⎛ i ⎞ e ∑ ⎜ ⎟ ix s = 0 s !( n − s )! ⎝ 2 x ⎠

s

[A5.27]

whence: eix ix eix ⎛ i⎞ h1(1) ( x) = − ⎜ 1 + ⎟ x ⎝ x⎠ ix ie ⎛ 3i 3 ⎞ h2(1) ( x) = ⎜1 + − 2 ⎟ x ⎝ x x ⎠ h0(1) ( x) =

and similarly for the spherical Hankel functions hn(2) ( x ) .

[A5.28]

Appendix A6

Specific impedances of several substances The following table presents data pertaining to common gases, liquids and solids, which will be useful for acoustical computations – in particular the values of sp the specific impedance Z ( ) = ρ 0 c0 . Table A6.1. Properties of various gases, liquids and solids

ρ 0 (kgm–3)

c0 (ms–1)

1.21

343

415

1.43

317

453

1.98

258

511

0.09

1270

114

0.6

405

242

Pure water (20 )

998

1481

1.5 106

Sea water (20o)

1026

1521

1.6 106

Alcohol ethyl

789

1159

0.9 106

SAE30 oil

920

1290

1.2 106

Mercury

13600

1451

19.7 106

Glycerine

1231

1895

2.3 106

Aluminum

2700

6300

17.0 106

Brass

8500

4700

40.0 106

Silver

10500

3700

38.9 106

Steel

7700

6100

47.0 106

Glass

2300

5600

12.9 106

Substance Air (20o) o

Oxygen (0 ) o

Carbon dioxide (0 ) o

Hydrogen (0 ) o

Steam (100 ) o

Z(

sp )

(rayl)

740


Substance

ρ 0 (kgm–3)

c0 (ms–1)

Concrete

2600

3100

8.1 106

Oak

720

4000

2.9 106

1100 / 950 / 1000

2400 / 1050 / 1550

2.6 / 1.0 / 1.6 106

Rubber (hard/soft/rho-c)

Z(

sp )

(rayl)

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[SIG 06] SIGRIST J.F., Symmetric and non-symmetric formulations for fluid-structure interaction problems: reference test cases for numerical developments in commercial finite element code, Pressure Vessel and Piping, Vancouver, July 25–28, 2006. [SOM 50] SOMMERFELD A., Mechanics of Deformable Bodies, Academic Press INC. Publishers, 1950. [SOU 00] SOULI M., OUAHSINE A., LEVINE L., ALE Formulation for Fluid-Structure Interaction Problems. Computer Methods in Applied Mechanics Engineering, 190, 659–675, 2000. [STA 70] STAKGOLD I., Boundary Value Problems of Mathematical Physics, Vol. 1 and 2, the Macmillan Company, 1970. [STO 51] STOKES G.G., On the effect of the internal friction of fluids on the motion of pendulums, Trans Cam. Phil. Soc. 9, pt. II, p. 8, 1851. [TEM 01] TEMKIN S., Elements of Acoustics, John Wiley &Sons 1976, Acoustical Society of America, 2001. [THO 53] THORNE, R.C., Multipole expansion in the theory of surface waves, Proc. Cambridge Phil. Soc. 49, pp. 707–716, 1953. [TUR 98] TURKEL E., YEFET A., Absorbing PML boundary layers for wave-like equations, Applied Numerical Mathematics, 27 pp. 533–557, 1998. [VAY 02] VAY J.-L., Asymmetric Perfectly Matched Layer for the Absorption of Waves, Jour. Comp. Phys. 183, pp. 367–399, 2002. [WAR 80] WARSI Z.U.A., Conservation Form of the Navier-Stokes Equations in General Non-Steady Coordinates, American Institute of Aeronautics and Astronautics Journal, 19 (2), pp. 240–242, 1980. [WATS 95] WATSON G.N., A Treatise on the Theory of Bessel Functions, Cambridge Mathematical Library, 1995. [WEH 60] WEHAUSEN J.V. and LAITONE V., Surface waves, in S. Flügge, editor, Encyclopedia of Physics, Vol. IX, pp. 446–778, Springer Verlag, 1960. [WEH 71] WEHAUSEN J.V., The motion of floating bodies, Annual Review of Fluid Mechanics 3, 237–268, 1971. [YEU 82] YEUNG R.W., The transient heaving motion of floating cylinders, J. Engineering Mathematics 16 97–119, 1982. [ZIE 89] ZIENKIEVICZ O.C. and TAYLOR R.L., The Finite Element Method. Basic Formulation and Linear Problems. 4th edn. Mac Graw Hill, 1989.

Index

1D wave equation 249 3D problems 120 cylindrical shell of low aspect ratio 122 immersed sphere 129 inverted pendulum 131 plate immersed in a liquid layer 120 3D wave equation boundary conditions 359, 363 in terms of displacement field 358 in terms of pressure 362 acoustical isolation of pipe systems 310 cavity in derivation to the main circuit 316 cavity inserted in the main circuit 311 acoustical modes analytical examples 364 cylindrical enclosure 369 direct orthogonality 362 displacement mode shapes 359 eigenvalue formulation 359 formulation 364, 369, 378 homogeneous fluids 362 modal series 361 orthogonality conditions 360 pressure mode shapes 363 rectangular enclosure 364 spherical enclosure 378 tube terminated by a pressure and a volume velocity nodes 264 tube terminated by two elastic impedances 264 tube terminated by two pressure nodes 263 tube terminated by two volume velocity nodes 263 uniform tube 262 acoustical resonances 9 acoustical sources concentrated 296

pressure 299 volume velocity 296 acoustics 1D wave equation 249 3D wave equation 245 and modal synthesis 245 and vibroacoustical interaction 245 linearized equations 245 added mass 4 breathing spherical shell 70 coefficient 4 continuous systems 81 coupled coaxial shells 95, 98, 109 cylindrical shell 90 cylindrical shell of low aspect ratio 125 flexible coaxial shells 219 flexible rectangular plate 101 free surface effect 222 immersed sphere 130 inverted pendulum 134, 137 matrix 48, 79 modal projection method 81, 83 nonlinear inertia 66 partly immersed rod 105 plate immersed in a liquid layer 121 rigid rectangular plate 102 system with 2 degrees of freedom 79 tube of variable cross-section 55 tube with a free atmosphere hole 58 water tank with flexible walls 119 anechoic boundaries 608 apparent phase speed 395 Arbitrary Lagrangian Eulerian (ALE) 3 audible frequency range 244 baffled circular piston directivity of the radiated sound 436 far field approximation 436

Index interference pattern 434 propagation in 3D space 432 radiation pattern 437 bending stiffness coefficient 649 Bernoulli equation 33 Bessel equation 397 spherical 272 Bessel functions 123, 371 orthogonality 376 series of 376 Bessel horns 272 conical 272 beta functions 187 boat wake 166 boundary conditions 25, 41 acoustical 359, 363 complex impedance 261 elastic impedance 259 fixed wall 47 free surface 47 general type 256 inertial impedance 260 linearized 28–39 pressure node 258 radiative damping 261 terminal impedances 256 volume velocity node 258 boundary element methods (BEM) 456 versus FEM 456 boundary layer 676 breakwater 172 bubble expansion 73 energy 73 phase portrait 74 pressure field in the liquid 75 bulk viscosity coefficient 706 buoyancy of a boat 225 antiresonant absorber 239 centre of buoyancy 226, 233 modal frequencies 228, 237 rolling induced by the swell 238 ship with rectangular cross-section 233 static equilibrium 226 CASTEM2000 coupled coaxial shells 108 vibroacoustic coupling 464 capillary waves 179 formulation 179 cavitation 1D model 554 1D vibroacoustic example 556 germs 21 pipes and ducts 554 pressure of 20

vibroacoustic coupling 554 cavitation bubbles 181–204 collapse 186 critical point 186 equilibrium pressure 182 minimum collapse radius 189 natural frequency of the breathing mode 193 nonlinear oscillations 191 oscillations 189 phase portrait 190 potential energy 182 Rayleigh collapse time 187 Rayleigh-Plesset equation 192 reference radius 182 static equilibrium 181 total energy 189 circular piston baffled 432 unbaffled 457 circular ring 86 circular tank sloshing frequencies 214 sloshing mode shapes 215 complex modes canonical form 601 chain of oscillators 589 physical interpretation 598 pipe systems 598 variable phase 598 complex sound velocity 600 compressible fluid plane waves 697, 699 plane waves in a tube 699 concentrated acoustical sources transfer matrix method 299 confined damped waves complex modal frequencies 699 confined fluid between two flexible plates 682 compressible 699 cylindrical annular gap 690 incompressible 677, 682, 687, 690, 695 multisupported tubes 695 piston-fluid systems 677 plane waves 699 plane waves in a tube 699 rigid vibrating plate 687 conical pipe 271 harmonic sequence 277 modal frequencies 274 mode shapes 274 open at both ends 273 stopped at one end and open at the other 273

749

750

Index

Couette’s flow 22 coupled coaxial shells added mass coefficients 95, 98 mode shapes 96, 107 coupled enclosures inertia coupling 407 modal equation 408 modal expansion method 406 stiffness coupling 407 coupled flexible plates added mass 685 viscous damping 686 coupling between sloshing and structural modes 216 flexible coaxial shells 217 formulation 216 cylindrical annular gap added mass 695 pressure field 694 velocity field 693 cylindrical enclosure acoustical formulation 369 acoustical modes 369 modal frequencies 371 modes of a cylindrical sector 377 mode shapes 371 cylindrical shell added mass coefficients 90 fluid-structure formulation 88 in infinite fluid 647 modal frequencies 87 mode shape 93 radiated power 650 radiated pressure 649 radiation 647 radiation damping 650 stiffness and mass coefficients 87 cylindrical shell of low aspect ratio 122 comparison of 3D and strip model 125 cylindrical vessels straight container 518 thermal expansion lyre 539 toroidal shell 531

damping matrices asymmetrical 594 gyroscopic systems 594 symmetrical 593 deep water waves 158 dispersive nature 158 space and time profiles 162 dimensionless parameters 39 compressibility effect 42 Froude number 41 gravity effect 41 inertial effect 39 oscillatory Mach number 42 oscillatory Reynolds number 44 Stokes number 44 surface tension effect 41 viscosity effect 43 Weber number 42 Dirac dipole 422 dispersion equation deep water waves 158 rectangular enclosure 365 shallow water waves 147 waves in a rectilinear canal 143 dispersion relation capillary waves 179 displacement potential 365 dissipation viscous 7 dissipative waves in a pipe analytical solution 612 modal solution 612 dynamical equations Euler equations 24 linearized fluid equations 26–28 mass conservation 14 modal 12 momentum equation 23 motion equations of a solid 10, 11 Navier-Stokes equations 24 Newtonian fluids 12–26 solid structures 10–12 vibration equation 11

d’Alembert’s paradox 134 damped acoustical modes pipe systems 595 terminal impedances 595 damped harmonic oscillator complex frequencies 583 energy decay 584 damped plane waves complex wave number 697 damped waves in a tube complex wave number 701 modal damping 701

elastic Lamé parameters 11 energy dissipation damping mechanisms 582 fluid friction 582 non-conservative coupling 582 radiation 582 viscous damping 582 equation of state 18 linearized 19 perfect gas 19 Euler equations 24 linearized 245

Index Eulerian viewpoint 3, 12 Euler-Lagrange equation 35 evanescent waves 270 expansion methods forced problems 497 formulation 497 modal truncature 505 seismic excitation 502 far acoustical field 412 Finite Element Method (FEM) CASTEM2000 108 coupled coaxial shells 108 fluid displacement formulation 564 fluid potential formulation 569 fluid pressure formulation 568 variational vibroacoustic formulation 564 versus BEM 456 vibroacoustic discretization 571 vibroacoustic Lagrangian 565, 568 vibroacoustic systems 463 FEM discretization 1D acoustic element 578 fluid displacement formulation 573 fluid potential formulation 578 fluid pressure formulation 575 vibroacoustic systems 571 flexible coaxial shells 217 added mass matrix 219 coupling parameter between sloshing and structure 219 large Froude numbers 221 resonant range 221 small Froude numbers 220 floating structures 225 antiresonant absorber 239 boundary value problem 670 buoyancy of a boat 225 centre of buoyancy 226, 233 energy loss ratio 669 energy radiated 669 equivalent viscous damping ratio 670 heave mode 668 modal frequencies 228, 237 radiation damping coefficient 669 rolling induced by the swell 238 ship with rectangular cross-section 233 static equilibrium 226 unsteady Bernoulli equation 671 unsteady Kirchhoff-Helmholtz equation 672 flow induced vibrations 10 flow rate mass flow versus volume velocity 49 fluid column model 48 1D equations 50

751

fluid confinement confinement ratio 40 fluid equations Bernoulli equation 33 dimensionless parameters 39 Euler equations 24 linearized 26–28 mass conservation 14 momentum equation 23 Navier equation 14 Navier-Stokes equations 24 fluid-elastic instability 26 fluid gap under a plate added mass 689 pressure field 689 velocity field 689 viscous damping 690 fluid-fluid interfaces loss 651 power reflection coefficient 652 power transmission coefficient 652 pressure reflection coefficient 651 pressure transmission coefficient 651 fluid layer approximation 98 coaxial cylindrical shells 98 other geometries 100 fluid-structure coupling 2, 7, 10 fluid-structure coupling 3D problems 120 between sloshing and structural modes 216 coupled coaxial cylindrical shells 94, 98 cylindrical shell 88 cylindrical shell with external fluid 92 cylindrical shell with internal fluid 84 flexible rectangular plate 100 fluid layer approximation 98 floating structures 225 free surface effects 139, 216–43 inertial 46, 47 rigid rectangular plate 101 strip model 85 fluid-wall-fluid interfaces far field transmission 667 finite 652 infinite 652 mass attenuation law 663 modal solution 667 sound transmission 661, 664 transmission loss 656 forced waves distributed velocity source over a surface 401 enclosures 399–411 equations 398 Green functions 399

752

Index

impulsive dipole source 405 impulsive monopole source 399 impulsive point velocity source 399 impulsive pressure source 405 local and far fields 412 moving strip of wall 403 piston-type source 402 rectangular enclosures 399 source terms 398 wave equation with source terms 399 waveguides 411 free surface problems boundary conditions 139 formulation 139 frequency spectrum diagram 391 Fresnel integrals 162 Froude number 41, 140 sloshing 220 gamma functions 187 Gibbs oscillations 176 gravity waves 6, 7 deep water 158 formulation 141 impacting a rigid wall 172 Kelvin wedge 166 rectilinear canal 140 shallow water 147 solitary waves 168 solitons 168 tidal waves 149 tsunamis 149 Green function 393, 405 and transfer functions 400 baffled 426 gravity waves in deep water 164 propagation in 3D space 421 spherical wave 631 Green identity 83 group velocity capillary waves 180 waves in a rectilinear canal 142 guided wave modes 387–398 as a combination of a wave-pair 393 characteristic length of decrease 390 cut-off frequency 390, 397 cylindrical waveguides 396 dispersion equation 389, 397 dispersive nature of non-plane waves 390 evanescent waves 390 formulation 388, 397 forward and backward guided waves 389 frequency spectrum 391 geometrical accidents 393 group velocity 392 in solids 387

phase velocity 389 physical interpretation 393 progressive waves 390 propagation of wave energy 396 rectangular waveguides 388 travelling waves 390 heat conduction damping attenuation length 705 entropy change 702 sound wave 704 thermal wave 704 thermoacoustic equations 702 thermoacoustic plane waves 703 wave equation 705 heat exchangers loose tubes 695 Helmholtz resonator case of a short neck-tube 295 higher plane wave modes 295 transfer matrix model 291 horns 269 Bessel 272 catenoidal 271 conical 271 horn function 269 Salmon 271 Webster equation 269 immersed sphere 129 impedance and mode coupling in waveguides 417 boundary conditions 256 change 250 complex 261 dimensionless 257 elastic impedance 259 general type 256 inertial impedance 260 pressure node 258 radiative damping 261 reflected and transmitted waves 250 specific 250 surface in waveguides 417 terminal impedances 256 transmission coefficient 250 uniform tube 249, 250 volume velocity node 258 incompressible fluid between two flexible plates 682 cylindrical annular gap 690 piston-fluid systems 677 rigid vibrating plate 687 inertial coupling continuous systems 81 directional aspect 83

Index discrete systems 48 fluid column model 48, 50 formulation 46, 47 infinite fluid cylindrical shell 647 oscillating sphere 643 pulsating sphere 629 inverted pendulum 131 inviscid fluid model 24 Kelvin wedge 166 Kirchhoff-Helmholtz integral open tube radiation 624 pulsating sphere 638 Kirchhoff-Helmholtz integral equation (KH) 439, 441 3D external and internal problems 452 boundary element methods (BEM) 456 low frequency range 457 nature of surface sources 456 plane acoustic waves triggered by a transient 446 plane waves 441 response in spectral domain 446 sluice gate 446 time interval of integration 444 unbaffled circular piston 457 volume contribution 445 Korteweg and de Vries equation (KdV) 168 dispersion relation 169 linear form 169 nonlinear solution 171 Lagrange multiplier 16, 35, 208 Lagrangian constrained 16, 35 superficial fluid 31 Lagrangian viewpoint 3, 12 laminar flow 25 Laplace capillary law 37 Laplacian curvilinear coordinates 84 cylindrical coordinates 85 Legendre equation 380 Legendre functions 380 associated 380, 385 Legendre polynomials 381 linearized boundary conditions 28–39 free surface in a gravity field 29 surface tension at fluid interfaces 34 wetted wall 28 linearized fluid equations 26–28 Euler equations 27 inertial coupling 46, 47 linearization procedure 26 wave equation 28

local acoustical field 412 Lommel integrals 373 Love equations 86 Mach number oscillatory 42, 50, 70, 362 membrane equation 657 fluid coupled 657 vibroacoustic travelling waves 660 modal coordinate system 362 modal density 366 Schroeder frequency 368 modal expansion method modes of coupled enclosures 406 modal projection method added mass 81 modal series 361 modal truncation stiffness coefficient criterium for selection 474, 477, 525 mode de pilonnement 231 mode shapes coupled coaxial shells 107 in-phase and out-of-phase 112 modification by fluid inertia 103 partly immersed rod 104 water tank with flexible walls 114 momentum flux tensor 24 multi degree of freedom systems chain of coupled oscillators 586 first-order formulation 586 non proportional damping 589 proportional damping 592 musical acoustics 244 musical drum vibroacoustic eigenproblem 551 vibroacoustic formulation 549 volume-dependent coupling 551 zero frequency mode 553 musical instruments brass 267 clarinet 268 drum 9 oboe 269 open and stopped pipes 268 pipe organs 267 recorder 273 saxophone 269 tempered scale 267 wind instruments 267 woodwind 267 Navier equation 14 Navier-Stokes equations 24 nonlinearity 25

753

754

Index

near and far fields pulsating sphere 635 Newtonian fluids isotropic 23 non reflecting boundaries 608 open end impedance conservative and dissipative 627 open tube radiation complex impedance 627 dimensionless impedance 628 Kirchhoff-Helmholtz integral 624 resultant force 626 terminal impedance 628 optimisation problems 35 and Lagrange multipliers 35 Dido problem 34 isoperimetric problem 34 oscillating sphere in infinite fluid 643 radiated power 644 radiated pressure 644 radiation 643 radiation damping 647 oscillatory Mach Number 50, 70 other damping mechanisms heat conduction 701 relaxation 706 thermoacoustic coupling 701 pendulum 7 added mass 8 buoyancy 8 damping 8 phase velocity capillary waves 180 waves in a rectilinear canal 142 physical properties air 19 seawater 20 pipe systems acoustical damping 596, 597 acoustical isolation 310 added mass coefficient 509 added mass matrix 510 anechoic impedance 609 cavitation 554 cavity in derivation to the main circuit 316 cavity inserted in the main circuit 311 change in cross-section 510 complex modal frequencies 596, 597 complex mode shapes 596, 597 coupling at bends 512 coupling at junctions 514 damped acoustical modes 595 dissipative fluid 608 dissipative waves 612

effect of end dissipation 606 equivalent sound velocity 516 horns 269 plane wave approximation 248 plane wave equations 248 plane wave model 506 potential fluid formulation 516 pressure fluid formulation 516 simplifications 506 speed of sound 329–352 structure formulation 515, 516 terminal impedances 595, 608 transverse coupling 508 vibroacoustic coupling 507 vibroacoustic formulation 515 pipes and tubes damped transfer matrix 599 piston-fluid systems 51–68 boundary conditions 483 conservative outlet 618 corrective length of a hole 58 damped waves 616 dissipative outlet 620 energy transfer 491 forced problem 486 harmonic force 486 inertial impedance at a hole 57, 59 low order expansion 484 modal expansion 471, 477, 481 nonlinear inertia 66 potential formulation 581 pressure formulation 466, 477 radiated sound power 621 rheonomic constraint 482 terminal impedance 624 transient force 490 uniform tube 51 with two degrees of freedom 76 zero frequency mode 481 Planck constant 271 plane damped waves attenuation coefficient 698 attenuation length 698 plane wave approximation pipe systems 248 plane waves equations in a pipe 248 forced by a pressure source 299 forced by a volume velocity source 298 plate bending stiffness coefficient 654 equation 654 modal frequencies 654 modal response 656 mode shapes 654 tensioned 654

Index plate immersed in a liquid layer 120 polytropic index 19 law 19, 334 potential flows 25 pressure concept 15, 16 pressure source 299 acoustical equations 299 forced wave equation equations 299 principle of least action 31 propagation in 3D space 421 1D Green function 430 2D Green function 428 baffled circular piston 432 baffled green functions 426 bounded by a fixed plane 424 bounded by a pressure nodal plane 426 cylindrical waves 427 dipole radiation 437 dipole sources 426 distributed monopole sources 427, 429 far field case 425 Green function 422 image source method 424 instantaneous acoustic intensity 423 Kirchhoff-Helmholtz integral equation (KH) 437, 439, 441 line source 428 mean acoustic intensity 423 monopole sources distributed within a surface 432 monopole sources distributed within a volume 431 plane waves 429 Rayleigh integral 432 unbaffled circular piston 437, 457 unbounded medium 421 weighted integral formulations 439 pulsating sphere added mass 631 comparison with viscous model 634 compressibility factor 632 converging wave 630 far field 635 fluid force 631 impedance 632 in infinite fluid 629 Kirchhoff-Helmholtz integral 638, 640 mean sound power 637 modal damping 632 monopole source 631 near field 635 outgoing wave 630 radiated energy 636

radiation 629 radiation damping 631 Rayleigh integral 638 quantum mechanics 270 matter wave 270 Planck constant 271 radiated power 613 radiation cutoff frequency 658 cylindrical shell 647 oscillating sphere 643 pulsating sphere 629 surface waves 668 radiative damping 6 rate of strains tensor 22 Rayleigh distance 434 Rayleigh integral baffled circular piston 432 monopole sources distributed within a surface 432 Rayleigh-Plesset equation 192 forced 194 rectangular bassin 209 rectangular enclosure acoustical formulation 364 acoustical modes 364 acoustical mode shapes 365 dispersion equation 365 modal frequencies 366 rectangular enclosures forced waves 399 rectangular plate flexible 100 rigid 101 rectangular tank deep pool sloshing frequencies 210 shallow pool sloshing frequencies 210 sloshing frequencies 210 sloshing mode shapes 210 reflected waves 250, 395 interface separating two media 354 tube with change of cross-section 250 tube with three propagating media 252 reflection coefficient oblique incidence 357 power 652 pressure 651 relaxation and molecular processes 706 Reynolds number oscillatory 44 Rodrigues formula 381 root finding bisection algorithm 195

755

756

Index

Newton algorithm 195 quasi-Newton algorithms 196 root mean square value 487 Saint Venant principle 421 in acoustics 421 in solids 421 Schroeder frequency 368 Schrödinger equation 269 tunnelling effect 271 second coefficient of viscosity 706 shallow water waves 147 non-dispersive nature 147 space and time profiles 160 shear modulus 11 shells coaxial 94, 98 cylindrical 84, 92 Love equations 86 sloshing modes 7, 140, 205–216 discrete systems 205 interconnected tanks 207 rectangular tank 209 U tube 205 sluice gate 446 water hammer 450 solitary waves 168 Korteweg and de Vries equation (KdV) 168 solitons 168 sound intensity 613, 614 mean 615 sound perception 244 threshold of pain 244 threshold of perception 244 sound power 615 sound propagation adiabatic 702 isothermal 702 sound transmission 651 mass attenuation law 663 sound velocity adiabatic 702 complex 600 isothermal 702 sound waves 9 source terms pressure source 398 specific impedance 250 spectral power density 487 speed of sound 18, 329–352 adiabatic 334 and fluid compressibility 329 bubble liquid 339 fluid contained within elastic walls 349 in air 338

in water 338 isothermal 334 mass-spring analogy 330 polytropic law 334 root mean square velocity of gas particles 333 sonorous line 330 state and temperature dependence 331 table of values 338 thermal conduction 334 two-phase mixture 339, 341 sphere oscillating rectilinearly 643 pulsating 629 spherical enclosure acoustical formulation 378 acoustical modes 378 case of a pressure nodal surface 386 modal frequencies 382, 386 mode shapes 382, 386 spherical functions 380 associated 380 spherical systems 68–76 breathing mode of an immersed spherical shell 68 early stage of a submarine explosion 71 stationary phase method 160 Stokes 7 Stokes number 44, 679, 685 confined waves 699 travelling waves 699 straight cylindrical vessel modal truncation 524 numerical aspects 524 numerical results 519 vibroacoustic displacement formulation 518 vibroacoustic pressure formulation 525 zero frequency mode 528 strain-stress relationship 16 stress tensor Cauchy 11 hydrostatic 15 strip model 85 Struve function 626 substantial derivative 12 convective rate of change 13 local rate of change 13 surface tension 6, 34, 179–204 capillary force 36 capillary length 37, 139 capillary waves 179 coefficient 6 experiment 34 Laplace capillary law 37 meniscus 38

Index surface waves 5 radiation 668 tank sloshing 209 tempered scale 267 eight-tone 267 half-tone 267 octave 267 quarter-tone 267 tone 267 terminal impedances numerical simulation 608 thermal conduction 334 thermal expansion lyre compressible formulation 543 incompressible formulation 539 transfer function 544 tidal waves 149 time-step integration explicit versus implicit 196 Newmark implicit method 194 TMM computational procedures assembled matrix equation 321, 323, 324 example application 327 external sources 323 general formulation for forced systems 320 impedance elements 323 isolation of a forced flow loop 327 multi-branched circuits 325 single branched circuits 324 toroidal shell Fourier expansion 534 vibroacoustic formulation, 532 transfer functions 400, 405 transfer function method (TMM) concentrated acoustical sources 299 tube with concentrated acoustical sources 301 transfer matrix damped 600 inverse 281 lack of symmetry 280 modal analysis 281 uniform tube element 279 transfer matrix method (TMM) advantages 282 assembling elements 282 cavity in derivation to the main circuit 316 cavity inserted in the main circuit 311 transfer matrix 279 two tube-elements with distinct crosssections 284 two tube-sections filled with distinct fluids 286

transformation adiabatic 19 isothermal 19 transmission coefficient 250 oblique incidence 357 power 652 pressure 651 transmission loss 652 transmitted waves 250 interface separating two media 354 tube with change of cross-section 250 tube with three propagating media 252 trial function 365 tsunamis 149–58 meteorological 152 nonlinear effects 156 seismic 149 travelling waves 270 tube with change in cross-sections 250 tube with concentrated acoustical sources modal expansion method 309 transfer matrix method 301 tube with three media radiated power 254 tube with three propagating media 252 tubular systems added mass coefficient 509 added mass matrix 510 cavitation 554 change in cross-section 510 coupling at bends 512 coupling at junctions 514 equivalent sound velocity 516 plane wave model 506 potential fluid formulation 516 pressure fluid formulation 516 simplifications 506 structure formulation 515, 516 transverse coupling 508 vibroacoustic coupling 507 vibroacoustic formulation 515 turbulent flow 25 two-phase mixture bubble vibrations 345 dispersive model 345 frequency range effect 348 homogeneous density 340 homogeneous Young modulus 340 isothermal gas behaviour 344 phase velocity 348 speed of sound 341 void fraction 339 wave equation 347

757

758

Index

unbaffled circular piston dipole radiation 437 Kirchhoff-Helmholtz integral equation (KH) 437 propagation in 3D space 437, 457 unconfined fluid compressible 697 plane waves 697 uniform tube acoustical equation 249 acoustical modes 262 impedance 249 terminated by a pressure and a volume velocity nodes 264 terminated by two elastic impedances 264 terminated by two pressure nodes 263 terminated by two volume velocity nodes 263 travelling waves 249 universal gas constant 19 velocity potential 25, 365 vibroacoustic coupling CASTEM2000 464 cavitation 554 contact condition 463 cylindrical vessel 518 displacement symmetrical formulation 463 energy transfer 491 expansion methods 497 forced problem 486, 497 harmonic force 486 holonomic constraint 567 mixed non symmetrical formulation 462 mixed symmetrical formulation 464 modal truncation stiffness coefficient 472, 474, 477, 525 modal truncature 505 penalty contact factor 464, 472, 564 pipe systems 515 plane wave formulation 515 simplified musical drum 548 straight vessel 518 symmetrization variable 464 thermal expansion lyre 539 toroidal shell 531 transient force 490 transient pressure source 493 tubular systems 515 vibroacoustic modes 9 vibroacoustic systems Finite Element Method (FEM) 563 viscosity coefficients 22 concept 21 dynamic viscosity 22, 23

kinematic viscosity 23 Newton fluid friction law 22 viscosity damping incompressible fluid 686 plates enclosing fluid 686 viscosity dissipation Stokes stress tensor 684 viscosity dissipation compressible fluid 697 incompressible fluid 677 viscous damping 680, 681 viscous damping complex mode shapes 589, 594 damping matrices 593 forced waves 604 harmonic oscillator 583 multi degree of freedom systems 585 peak width 585 viscous shear waves boundary layer 676 viscous shear waves 673 evanescent 676 Stokes second problem 674 volume velocity source 296 acoustical equations 297 forced wave equation equations 298 vorticity vector 24 water hammer 450, 554 water tank with flexible walls added mass 119 mode shapes 114 waves evanescent 270 group velocity 142 instantaneous intensity 146 mean intensity 146 phase velocity 142 power 146 propagation delay 142 space and time profiles 159 reflected 250 transmission coefficient 250 transmitted 250 travelling 270 wave impacting a rigid wall 172 equivalent mass 173 pressure impulse 173 spatial profile 177 wave plane 395 wave vector 365 waveguides characteristic length of decrease 390 cut-off frequency 390, 397 cylindrical 396 dispersion equation 389, 397

Index dispersive nature of non-plane waves 390 evanescent waves 390 excitation by a vibrating membrane 412 formulation 388, 397 forward and backward guided waves 389 frequency spectrum 391 geometrical accidents 393 group velocity 392 impedance surface 417 local and far forced fields 412 mode coupling at impedance changes 417 phase velocity 389 physical interpretation 393 progressive waves 390

propagation of wave energy 396 rectangular 388 travelling waves 390 Weber number 42 weighted integral formulations 439 Kirchhoff-Helmholtz integral equation (KH) 439, 441 weighting functional vector 439 Young modulus of a fluid 17 zero frequency modes 465 piston-fluid systems 481 straight cylindrical vessel, 528 toroidal shell 534

759


Colour plates

The following colour plates refer to topics covered in the text. They result from computations programmed using the software MATLAB. Colour scales are used as a convenient manner to visualize the field variables – pressure and displacement (or one of its derivatives) – within the fluid domain. Also, lines corresponding to isovalues of the pressure field and arrows pertaining to the displacement or velocity field are sometimes also shown. Salient qualitative features of these results, to illustrate interesting points, are discussed in the text.

n =1

n=2

n=3

n=7

Plate 1. Oscillations of an incompressible fluid inside a cylindrical vibrating shell, for different shell mode shapes: the colours pertain to the pressure field and the arrows to the acceleration field

n =1

n=2

n=3

n=7

Plate 2. Oscillations of an incompressible fluid enclosing a cylindrical vibrating shell, for different shell mode shapes: the colours pertain to the pressure field and the arrows to the acceleration field

n =1, m = 0

n =1, m =1 Plate 3. Sloshing fluid motions on a rectangular pool for two low-order mode shapes of the free surface: the coloured lines pertain to isovalues of the pressure field and the black arrows display the flow acceleration field

n =1, m = 2

n =3,m = 2 Plate 4. Sloshing fluid motions on a rectangular pool for two higher-order mode shapes of the free surface: the coloured lines pertain to isovalues of the pressure field and the black arrows display the flow acceleration field

Plate 5. The lowest-frequency spherically symmetric and axisymmetric acoustic modes inside spheres of radius R = 1 m, respectively with a rigid wall (upper plots) and with a zeropressure boundary (lower plots), for sound velocity c f = 243 m/s

Plate 6. Axisymmetric acoustic modes ( m = 0 ) inside a spherical volume of radius R = 1 m enclosed by a rigid wall, with sound velocity c f = 243 m/s , for two values of the azimuthal index n = 1, 2 and two values of the radial index l = 1, 2

Plate 7. Non-axisymmetric acoustic modes inside a spherical volume of radius R = 1 m enclosed by a rigid wall, with sound velocity c f = 243 m/s , for increasing values of the azimuthal index m = 1 ~ 4 and constant values n = 4 and l = 1

Plate 8. Guided wave mode of order (1, 0) in a rectangular waveguide of width 2b , represented as a sum of two planar waves travelling with oblique incidence ±θ , for three values of the frequency ratio c f 2bf = λ 2b

Plate 9. Guided wave mode of order (3, 0) in a rectangular waveguide of width 2b , represented as a sum of two planar waves travelling with oblique incidence ±θ , for three values of the frequency ratio c f 2bf = λ 2b

Plate 10. Modes of the decoupled membrane and enclosure for an enclosure height Lz = 0.2 m

Plate 11. Vibroacoustic coupled modes for an enclosure height Lz = 0.2 m

Plate 12. Modes of the decoupled membrane and enclosure for an enclosure height Lz = 0.7 m

Plate 13. Vibroacoustic coupled modes for an enclosure height Lz = 0.7 m

Modelling of Mechanical Systems: Fluid-Structure Interaction, Volume 3 (Modelling of Mechanical Systems)

Modelling of Mechanical Systems: Discrete Systems, Volume 1 (Modelling of Mechanical Systems) (Modelling of Mechanical Systems)

Modelling of Mechanical Systems: Discrete Systems, Volume 1 (Modelling of Mechanical Systems) (Modelling of Mechanical Systems)

Modelling of Mechanical Systems: Structural Elements, Volume 2 (Modelling of Mechanical Systems)

Modelling of Mechanical Systems: Structural Elements, Volume 2

Impacts in mechanical systems: analysis and modelling

Dynamics of Mechanical Systems

Synchronization of Mechanical Systems

Dynamics of mechanical systems

Vibration of Mechanical Systems

Modelling of Marine Systems

Fundamentals of Robotic Mechanical Systems

Geometric control of mechanical systems

Fundamentals of Robotic Mechanical Systems

Fundamentals of Robotic Mechanical Systems

Opto-mechanical Systems Design

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Impacts in Mechanical Systems

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Mathematica Mechanical Systems

BOAT MECHANICAL SYSTEMS HANDBOOK

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Modelling Forest Systems

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Identification and Control of Mechanical Systems

Modelling of Mechanical Systems: Fluid-Structure Interaction, Volume 3 (Modelling of Mechanical Systems)

Modelling of Mechanical Systems: Discrete Systems, Volume 1 (Modelling of Mechanical Systems) (Modelling of Mechanical Systems)

Modelling of Mechanical Systems: Discrete Systems, Volume 1 (Modelling of Mechanical Systems) (Modelling of Mechanical Systems)

Modelling of Mechanical Systems: Structural Elements, Volume 2 (Modelling of Mechanical Systems)

Modelling of Mechanical Systems: Structural Elements, Volume 2

Impacts in mechanical systems: analysis and modelling

Dynamics of Mechanical Systems

Synchronization of Mechanical Systems

Dynamics of mechanical systems

Vibration of Mechanical Systems

Modelling of Marine Systems

Fundamentals of Robotic Mechanical Systems

Geometric control of mechanical systems

Fundamentals of Robotic Mechanical Systems

Fundamentals of Robotic Mechanical Systems

Opto-mechanical Systems Design

Mechanical engineering systems

Impacts in Mechanical Systems

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Mathematica Mechanical Systems

BOAT MECHANICAL SYSTEMS HANDBOOK

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Modelling Forest Systems

Modelling Diesel Combustion (Mechanical Engineering Series)

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Dynamics and Control of Hybrid Mechanical Systems

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