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) ....................................................... 147 3.2.4. Application of the shallow wave theory to tsunamis......................... 149 3.2.4.1. Seismic tsunami waves ...................................................... 149 3.2.4.2. Meteorological tsunamis.................................................... 152 3.2.4.3. Nonlinear limitation of resonant waves ............................. 156 3.2.5. Deep 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
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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
Introduction to fluid-structure 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
Fluid-structure interaction
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
Introduction to fluid-structure coupling
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
Fluid-structure interaction
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,
Introduction to fluid-structure coupling
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
Fluid-structure interaction
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
Introduction to fluid-structure coupling
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
Fluid-structure interaction
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
Introduction to fluid-structure coupling
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
Fluid-structure interaction
[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:
Introduction to fluid-structure coupling
⎞ 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
Fluid-structure interaction
)
⎞ ⎛∂ 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
Introduction to fluid-structure coupling
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
Fluid-structure interaction
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]
Introduction to fluid-structure coupling
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
Fluid-structure interaction
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
Introduction to fluid-structure coupling
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
Fluid-structure interaction
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
Introduction to fluid-structure coupling
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
Fluid-structure interaction
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
Introduction to fluid-structure coupling
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
Fluid-structure interaction
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:
Introduction to fluid-structure coupling
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
Fluid-structure interaction
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:
Introduction to fluid-structure coupling
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
Fluid-structure interaction
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 )
Introduction to fluid-structure coupling
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
Fluid-structure interaction
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
Introduction to fluid-structure coupling
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
Fluid-structure interaction
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]
Introduction to fluid-structure coupling
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
Fluid-structure interaction
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
Introduction to fluid-structure coupling
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
Fluid-structure interaction
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.
Introduction to fluid-structure coupling
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
Fluid-structure interaction
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]
Introduction to fluid-structure coupling
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
Fluid-structure interaction
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
Fluid-structure interaction
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
[1.104]
where α f is the capillary length already defined, see relation [1.91]. In the case of an air/water interface at 20°C, capillarity is negligible as soon as L is larger than a few millimetres. To compare directly surface tension with fluid inertia, it is convenient to express the condition [1.95] in terms of pressure solely, by using once more the momentum equation [1.68]. Discarding here the gravity term, we arrive at: ⎡ σ ⎛ ∂ 2p ∂ 2 p ⎞⎤ p(x, y,z) − 0 ⎜ 2 + ⎢ ⎟⎥ = 0 ρ 0 ⎝ ∂ x ∂z ∂ y 2 ∂z ⎠ ⎦ H ⎣
Once more the subscript ( 0 ) is used here instead of
[1.105]
( ) to specify that we refer to f
the value in still fluid of the associated quantity. Dimensional analysis of the condition [1.105] leads to define the oscillatory Weber number: We =
ρ0ω 2 L3 σ0
[1.106]
Clearly, in the range We >>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
Introduction to fluid-structure coupling
43
domain Λa << 1 , except of course in the special case of a fluid entirely enclosed by fixed walls, which could not move at all if compressibility had been neglected. Depending on the values of the scale lengths Lx , Ly , Lz , related to the geometry of the fluid volume, it often happens that compressibility is important in two or even a single direction solely, as will be shown in Chapters 4 and 5. 1.3.3.5 Fluid viscosity Fluid viscosity governs viscous friction and adherence of the fluid to the walls. To obtain a dimensionless number measuring the relative magnitude of the inertia to the viscous forces, it is suitable to start from the linearized Navier-Stokes equations [1.54], which are conveniently simplified here by assuming Λa << 1 and no external load: ρ 0 div X f = 0 [1.108] ρ 0 X f + grad p − μ0 ΔX f = 0 With the aid of the mass equation in conjunction with the following vector identity: [1.109] ΔX f = grad ⎡⎢ div X f ⎤⎥ − curl ⎡⎢curl X f ⎤⎥ ⎣ ⎦ ⎣ ⎦ The momentum equation can be transformed into: ρ 0 X f + grad p + μ0 curl ⎡⎢curl X f ⎤⎥ ⎣ ⎦ Then the following identities, div ⎢⎡curlX f ⎥⎤ = 0 ; Δ ⎡⎣ grad p ⎤⎦ = grad [ Δp ] ⎣ ⎦
[1.110]
[1.111]
allow one to eliminate either fluid displacement or pressure by taking either the divergence or the Laplacian of the momentum equation [1.110]. The following set of equations is obtained: Δp = 0 grad p.n
(W )
= − ρ 0 X s .n
ΔX f − ν 0 Δ 2 X f = 0 X f − X s =0 (W )
(W )
[1.112]
44
Fluid-structure interaction
According to the first equation [1.112], the fluctuating pressure is still governed by the incompressible law of an inviscid fluid and fluid displacement is governed by a viscous wave equation which will be further studied in Chapter 7. Dimensional analysis of the viscous wave equation leads to the definition of the oscillatory Reynolds number, often known also as the Stokes number: Sν =
ω L2 ν0
[1.113]
In the range Sν >> 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
Fluid-structure interaction
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
Fluid-structure interaction
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
Fluid-structure interaction
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
Fluid-structure interaction
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
Fluid-structure interaction
+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
Fluid-structure interaction
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
Fluid-structure interaction
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
Fluid-structure interaction
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
Fluid-structure interaction
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
Fluid-structure interaction
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
Fluid-structure interaction
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 << L , hence in most cases of practical interest. Finally, as an exercise the
reader can deduce the same results by using the Newtonian approach, starting from the nonlinear momentum equation projected along the axial direction. 2.2.3
Systems with spherical symmetry
2.2.3.1 Breathing mode of a spherical shell immersed in a liquid The geometry of the problem is depicted in Figure 2.14. The shell of radius R0 and thickness e vibrates in an infinite extent of liquid according to the breathing mode. So, the shell displacement U s is radial and uniform. Particularizing the Love equations (cf. [AXI 05], Chapter 8) to the present problem, it is readily found that, in vacuum, the modal equation can be written as:
Inertial coupling
US e
69
iωt
Figure 2.14. Spherical shell immersed in an infinite extent of liquid
⎛ 2 Es ⎞ 4π eR02 ⎜ U s − ω 2 ρ S R02U s ⎟ = 0 ⎝ 1 −ν s ⎠
[2.63]
leading to the natural frequency: f 0( ) = v
cs where cs = 2π R0
2 Es ρ s (1 − ν s )
[2.64]
where the superscript (v) stands for ‘in vacuum’. It is recalled that cs is the speed of the elastic dilatational waves in the spherical shell. Provided a displacement U s of unit magnitude is considered, the modal mass is equal to the physical mass: M s = 4π R02 eρ s
[2.65]
For a vibration of magnitude U s , the elastic membrane stresses are: N θθ = N ϕϕ = N =
Es e U s 1 − ν s R0
[2.66]
Denoting by N a the admissible stress up to which the material remains in the elastic range, the largest magnitude U a admissible to remain within that range is given by: U a (1 − ν s ) N a (1 − ν s ) σ a ≤ = R0 Es e Es
[2.67]
70
Fluid-structure interaction
As is well known, σ a / Es does not exceed a few 10−3 in most solids. Hence, discussion is restricted here to the realistic range U s << R0 . Due to the spherical symmetry, the fluid velocity field is also purely radial. Since compressibility is neglected, U f ( r ) is derived immediately by stating that the volume velocity through concentric spherical surfaces of radius R0 and r respectively, are the same. Hence: ⎛R ⎞ U f ( r ) = U s ⎜ 0 ⎟ ⎝ r ⎠
2
[2.68]
The kinetic energy of the liquid is then given by the following integral: ⌠
Eκ = 2πρ f U s2 R04 ⎮⎮
∞
⌡R0
dr 3 = 2πρ f R03U s2 = M f U s2 r2 2
⇒ M a = 3M f
[2.69]
M f is the mass of fluid which would fill the sphere.
The following comments can be made concerning such a result: 1. As the fluid extends to infinity, it can be questioned whether it can be reasonably modelled as incompressible, or not. This point will be further discussed in Chapter 5 in relation to spherical sound waves. It will be shown that the lengthscale relevant to the present problem is precisely the radius of the sphere. Hence, compressibility can be safely neglected, provided the oscillatory Mach Number ω R0 / c f defined in Chapter 1, subsection 1.3.3.4, is less than unity. 2. Though an infinite mass of fluid is set into motion by the shell vibration, the kinetic energy of the fluid remains finite because U f ( r ) decreases sufficiently fast with r. This would not be the case in a cylindrical geometry as detailed in subsection 2.3.2. Indeed, if a circular cylindrical shell of infinite length is immersed in an infinite extent of incompressible liquid, mass flow conservation leads to the radial fluid velocity: R U cyl ( r ) = U S 0 r
[2.70]
As a consequence, the added mass coefficient per unit length of the cylinder is found to be infinite. 3. Coming back to the case of the sphere, as could be anticipated, M a is related to the radius of the shell and to fluid density but differs from any physical volume of the system which could be defined a priori. Furthermore, the following simple expression for the added mass ratio is easily derived:
Inertial coupling
μf =
M a ⎛ ρ f ⎞ ⎛ R0 ⎞ =⎜ ⎟⎜ ⎟ M s ⎝ ρs ⎠ ⎝ e ⎠
71
[2.71]
Therefore, in the case of thin shells, large added mass effects can be expected, even if the fluid density is much less than that of the solid. As an illustration, let us consider the case of an aluminium shell R0 = 0.5m and e = 1mm immersed in water. The natural frequency of the breathing mode is found to be f 0( v ) 2.74 kHz in vacuum and f 0( w) 201Hz in water ( M a 1571kg , μ f 185 ). Finally, adopting the value cw 1500 m/s , the oscillating Mach number is found to be ω w R0 / cw 0.42 , indicating that the incompressibility assumption is still acceptable, at least as a first approximation. 2.2.3.2 Early stage of a submarine explosion As the calculation leading to [2.68] is based on a balance of mass flow, it holds also in the case of large displacements, provided calculation is based on the actual configuration R(t) instead on that of static equilibrium R0 , as already illustrated in subsection 2.2.2.6. The result becomes: ⎛R⎞ U f ( r ) = R ⎜ ⎟ ⎝r⎠
2
[2.72]
and the kinetic energy takes on the non quadratic form: ⌠ Eκ = 2πρ f R 2 R 4 ⎮⎮
∞
dr = 2πρ f R 3 R 2 2 ⌡R (t ) r
[2.73]
As a first application, we consider the expansion of a spherical bubble driven by the pressure of the internal gas. This corresponds typically to the early stage of the explosion of a submarine mine. Hereafter, the quantities referring to the gas are labelled by the subscript (G) and those referring to the liquid by (L). The gas elasticity is described by the polytropic law [1.31], written here as: ⎛R ⎞ PG = PG 0 ⎜ 0 ⎟ ⎝ R⎠
3γ
[2.74]
where the polytropic index is denoted γ instead of γ p to alleviate the notation. PG 0 is the gas pressure at the initial radius R0 and PG that at the current value R. The numerical applications, presented here for illustrative purposes only, were carried out by assuming γ = 1.4 , which corresponds to an adiabatic expansion of air. Finally, the hydrostatic pressure of water is PL . As a simplifying approximation
72
Fluid-structure interaction
which is consistent with a spherical expansion of the bubble, PL is considered to be constant. Furthermore, compressibility of water is disregarded. As a consequence, physical validity of the model is restricted to the early stage of the explosion such that PG >> 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 gas term is largely predominant; accordingly the slope of the
Inertial coupling
73
potential is negative and very steep, while 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 PG 0
[2.78]
It is of particular interest to notice that cb combines the elasticity of the gas and the inertia of the liquid. As will be evidenced just below, it characterizes the expansion rate of the bubble. Starting from [2.77], the law of motion is determined by the following integral: ⌠
t=
R0 cb
3 (γ − 1) ⎮⎮ ⎮ 2 ⎮
r (t )
dr
r
⎮ ⌡1
−3
−r
−3γ
+ α (γ − 1) ( r −3 − 1)
[2.79]
where r = R ( t ) / R0 is the dimensionless radius of the bubble. Singularity of the kernel at r = 1 is easily removed by expanding it to the first order in r − 1 = ε << 1 . This gives: ⌠
R t (ε ) = 0 cb
3 (γ − 1) ⎮⎮ 2 ⎮⎮⎮
ε
⌡0
dε 3 (γ − 1)(1 − α ) ε
=
R0 2ε cb 1 − α
[2.80]
Hence, numerical calculation of [2.79] can be easily performed on the computer. According to the conservative model, the motion is periodic, as indicated by the closed trajectories of the phase portrait of Figure 2.16, which correspond to a few values of PL ranging from 0 up to 100 bar. As could be expected, maximum expansion of the bubble is controlled by the hydrostatic pressure, diminishing significantly as PL increases while the early stage of motion up to the maximum velocity is controlled by the pressure within the bubble. A sample of time-histories of displacement and velocity during the expanding stage of motion is shown in Figure 2.17 and 2.18, respectively. The radius of the bubble is doubled in 4.4 ms.
74
Fluid-structure interaction
Maximum liquid velocity is of the same order of magnitude as the characteristic speed cb (about 316 m/s in the present example). The maximum speed is reached at a time of about 1.5 ms, its value increasing moderately if PL decreases, varying from about 200 m/s for PL = 100 bar to about 300 m/s for PL = 0 , which is substantially less than the speed of sound in water. A submarine explosion implying an initial pressure larger than about 22.5 kbar leads to supersonic velocities, hence to shock waves.
Figure 2.16. Phase portrait of the bubble
Figure 2.17. Time history of the bubble expansion
Inertial coupling
75
As an exercise, the pressure field induced in the liquid can be determined by using the nonlinear momentum equation: ⎛ ∂U L ∂U L ⎞ ∂P ρL ⎜ +UL =0 ⎟+ ∂r ⎠ ∂r ⎝ ∂t
[2.81]
where r denotes the radial position of a point within the liquid ( r ≥ R0 ). Substituting the velocity field [2.68] into [2.81], we can solve for the pressure. After a few elementary manipulations we arrive at the following results: 4 2 2 ⎛ ⎛ R ⎞2 ∂P + 2 RR − 2 R R = −ρL ⎜ ⎜ ⎟ R ⎜⎝ r ⎠ r2 r5 ∂r ⎝
⎞ ⎟⎟ ⇒ ⎠ 4 2 2 2 ⎛ RR RR R ⎛R⎞ ⎞ +2 − P(r; t ) = ρ L ⎜ ⎜ ⎟ ⎟ + PL ⎜ r r 2 ⎝ r ⎠ ⎟⎠ ⎝
[2.82]
The pressure at the gas-liquid interface is: ⎛ 3R 2 ⎞ P ( R; t ) = PL + ρ L ⎜ RR + ⎟ 2 ⎠ ⎝
[2.83]
Figure 2.18. Time-histories of the velocity
On the other hand, with the aid of relations [2.73] to [2.76], the Lagrange equation of the system is obtained as: ⎛ ⎛ ⎞ 3R 2 ⎞ 4π R 2 ⎜ ρ L ⎜ RR + ⎟ − ( PG − PL ) ⎟ = 0 2 ⎠ ⎝ ⎝ ⎠
[2.84]
76
Fluid-structure interaction
Equation [2.83] shows that the fluctuating pressure P( R; t ) − PL at the bubble boundary is precisely equal to that induced by the inertia of the liquid. Equation [2.84] shows that the inertial force is exactly balanced by the driving pressure force 4π R 2 ( PG (t ) − PL ) as should be. In Figure 2.19, the pressure within the bubble is plotted versus the radius in the case of an explosion taking place at 100 m depth. The driving pressure PG − PL is positive in the domain above the horizontal dashed line. Because of the term proportional to R 2 in the inertia force this includes a part of the range in which the acceleration is negative. The reader interested in the subject of submarine explosion can be referred to the textbook by Cole [COL 48].
Figure 2.19. Fluctuating gas pressure of the bubble
2.2.4
Piston-fluid system with two degrees of freedom
2.2.4.1 Natural modes of vibration The piping system sketched in Figure 2.20 is a T-junction connecting two straight tubes (AO) and (BC) containing an incompressible liquid, bounded at O by a free surface and at B and C by two mass-springs systems, assumed to be identical (mass M s , stiffness coefficient K s ), in order to simplify the algebra. On the other hand, A is the crossing point of the axes Ox and Oz of the tubes, which can be assumed to be orthogonal to each other without loss of generality. To simplify further the algebra, we choose BA = AC =OA/2 = L/2 and the cross-sectional area S f of the tube (OA) is twice that of the tube (BC) s f = S f / 2 . X 1 and X 2 are the displacements of the pistons. In the absence of fluid, they would vibrate independently from each other at the same natural pulsation:
Inertial coupling
ωs = K s / M s
77
[2.85]
Figure 2.20. Two mass-spring systems coupled by a liquid column enclosed in a T tube
Due to the incompressible liquid, we have to deal with a 2 DOF system coupled by the fluid inertia. Because of the symmetry with respect to the Oz axis, solution of the problem is immediate. The pair of coupled modes consists of one in-phase mode of shape [ϕ1 ] = [1 1] and one out-of-phase (phase opposition) mode of shape T
[ϕ 2 ]T = [1
−1] . Furthermore, it is easily seen that the in-phase mode induces no
liquid motion in the tube (OA). As a consequence, the corresponding added mass is the physical mass of the liquid contained in the tube (BC), which is written as: M a(1) = 2m f = 2 ρ f ( s f L / 2 ) = ρ f s f L
[2.86]
In contrast, the out-of-phase mode induces necessarily an oscillation of the liquid contained in the tube (OA), characterized by the fluctuating mass flow: ρ f q = 2 ρ f s f X = ρ f S f X , where X = X 1 = − X 2 .
[2.87]
Thus, due to the particular value of the cross-sectional ratio S f / s f = 2 , the fluctuating velocity has the same magnitude in each branch of the piping system and the added mass of the out-of-phase mode is readily found to be: M a( 2) = 6m f = 3ρ f s f L
[2.88]
78
Fluid-structure interaction
Accordingly, the natural pulsations of the coupled system are: 2K s
ω1 =
2M s + M a( ) 1
; ω2 =
2K s
2M s + M a(
[2.89]
2)
As an exercise, the problem is solved for a confinement ratio σ f = s f / S f , based first on the Lagrange equations and then on the Newtonian approach. The first method brings out that the fluid inertia can be characterized by an added mass matrix, which is symmetrical and positive definite, as suitable for any conservative system. The second method allows one to determine the modal pressure field associated with the vibration of the pistons. 2.2.4.2 Lagrange’s equations The elastic energy stored in the springs is: Ee =
1 K s ( X 12 + X 22 ) 2
[2.90]
The kinetic energy is first expressed in terms of the structural and fluid velocities in each tube separately: 1 2
M s (X 12 + X 22 ) + 12 m f (X 12 + X 22 ) + 12 M f X 32
[2.91]
where m f = ρ f s f L / 2; M f = ρ f S f L = 2m f / σ f .
The fluid velocity X 3 in the tube (OA) is related to X 1 and X 2 through the condition of mass conservation at the junction A, which implies: q1 + q2 + q3 = 0 q1 = + s f X 1 ; q2 = − s f X 2 ; q3 = − S f X 3
[2.92]
By convention, the volume velocity is assumed to be positive if directed toward A, that is if it enters into the T junction. Then, relation [2.92] is used as a holonomic condition to eliminate the superfluous variable: X 3 = σ f ( X 1 − X 2 )
[2.93]
Thus, the Lagrangian is written as the quadratic form: L=
1 2
(( M + (1 + 2σ ) m ) ( X s
f
f
2 1
+ X 22 ) − 4σ f m f X 1 X 2 − K s ( X 12 + X 22 )
and the Lagrange equations take on the matrix form:
)
[2.94]
Inertial coupling
⎡1 + 2σ f ⎡1 0⎤ ⎡ X 1 ⎤ ⎡ ⎡1 0⎤ + ⎢M s ⎢ + mf ⎢ Ks ⎢ ⎢ ⎥ ⎥ ⎥ ⎣0 1⎦ ⎣ X 2 ⎦ ⎢⎣ ⎣0 1⎦ ⎣ −2σ f
−2σ f ⎤ ⎤ ⎡ X1 ⎤ ⎡ 0⎤ = ⎥ 1 + 2σ f ⎥⎦ ⎥⎦ ⎢⎣ X 2 ⎥⎦ ⎢⎣ 0⎥⎦
79
[2.95]
In [2.95], we recognize the canonical form:
[ K ][ X ] + [ M ] ⎡⎣ X ⎤⎦ = [0]
[2.96]
which governs the free vibrations of a N-DOF conservative linear system. It is recalled that the structural stiffness matrix [ K s ] is symmetric positive, whereas the
structural mass matrix [ M s ] is symmetric, positive definite. Inertial coupling by the
fluid is accounted for by the so-called added mass matrix [ M a ] , which is also symmetric, positive and definite except in the very specific case of pure tangential motion of the wetted wall, as proved later in subsection 2.3.1. Consequently, the natural frequencies of the vibration modes are lowered with respect to the values in vacuum. Depending whether [ M a ] is diagonal or not, the system vibrates according to the same mode shapes as in vacuum, or not. In the present example, solution of the modal equation related to [2.95] is particularly simple, due to the symmetry of the physical system. As expected, the frequency of the in-phase mode is independent of the confinement ratio: f1 =
1 2π
Ks Ms + mf
In contrast, the frequency of the out-of-phase mode decreases as the confinement ratio increases: f2 =
1 2π
Ks M s + (1 + 4σ f ) m f
If σ f tends to zero (i.e. S f tends to infinity), the frequency of the out-of-phase mode tends asymptotically to that of the in-phase mode. As further clarified in the next subsection by computing the fluctuating pressure field, if the volume of the tube (OA) is sufficiently increased, the pressure within it is practically imposed by the boundary condition on the free surface, hence it becomes practically zero. As a consequence, the mass flow in (OA) also vanishes and f 2 tends to f1 . In contrast, if σ f tends to infinity (i.e. S f tends to zero), the out-of-phase mode disappears since
the related added mass coefficient tends to infinity, which means that the volume of the incompressible fluid in the tube (BC) cannot be varied when S f vanishes.
80
Fluid-structure interaction
2.2.4.3 Newtonian treatment of the problem Let us consider again a harmonic vibration of pulsation ω . The pistons are governed by the following equations:
( K δ ( x) − ω M δ ( x ) ) X = − ps δ ( x ) ( K δ ( L − x) − ω M δ ( L − x ) ) X = + ps δ ( L − x ) s s
2
1
s
2
s
f
2
[2.97]
f
In agreement with [2.92], the flow rates induced by the motion of the pistons are: q1 = iω s f X 1 ; q2 = −iω s f X 2 ; q3 = −iωσ f s f ( X 1 − X 2 )
[2.98]
The fluctuating pressure in each tube is governed by an equation of the type [2.16], producing here the system of two equations: d2 p − ω 2 ρ f ( X 1δ ( x ) + X 2δ ( L − x ) ) = 0 dx 2 d2 p − ω 2 ρ f σ f ( X1 − X 2 ) δ ( z ) = 0 dz 2
[2.99]
By using the same integration procedure as in [2.19], the following results are obtained: ⎛ dp ⎞ ⎛ dp ⎞ 2 2 ⎜ ⎟ = ω ρ f X 1 ; ⎜ ⎟ = −ω ρ f X 2 dx dx ⎝ ⎠B ⎝ ⎠C
⎛ dp ⎞ ; ⎜ ⎟ = ω 2 ρ f σ f ( X 1 − X 2 ) [2.100] ⎝ dz ⎠ A
The pressure field in the three branches of the T-junction follows immediately as: L⎞ L ⎛ ⎛L⎞ p1 ( x ) = +ω 2 ρ f X 1 ⎜ x − ⎟ + p1 ⎜ ⎟ 0 ≤ x ≤ 2⎠ 2 ⎝ ⎝2⎠ ⎛L ⎞ ⎛L⎞ L ≤x≤L p2 ( x ) = −ω 2 ρ f X 2 ⎜ − x ⎟ + p2 ⎜ ⎟ 2 ⎝ ⎠ ⎝2⎠ 2
[2.101]
p3 ( z ) = +ω 2 ρ f σ f ( X 1 − X 2 )( z − L )
Continuity of the pressure at A implies that: ⎛L⎞ ⎛L⎞ p1 ⎜ ⎟ = p2 ⎜ ⎟ = p3 ( 0 ) = −ω 2 ρ f σ f L ( X 1 − X 2 ) ⎝2⎠ ⎝2⎠
[2.102]
Substituting back [2.102] into [2.101], pressure is entirely determined as homogeneous linear functions of the accelerations of the pistons. The later are loaded by the pressure forces:
Inertial coupling
⎛ L ⎞ − s f p1 ( 0 ) = ω 2 ρ f s f ⎜ X 1 + σ f ( X 1 − X 2 ) L ⎟ = ω 2 m f (1 + 2σ f ) X 1 − 2σ f X 2 2 ⎝ ⎠ L ⎛ ⎞ + s f p2 ( L ) = ω 2 ρ f s f ⎜ X 2 − σ f ( X 1 − X 2 ) L ⎟ = ω 2 m f (1 + 2σ f ) X 2 − 2σ f X 1 2 ⎝ ⎠
(
)
(
)
81
[2.103]
Substituting the forces [2.103] into the mass-spring equations [2.97], the frequency or spectral version of the matrix equation [2.95] is recovered. Furthermore, the equations [2.98] and [2.102] clearly indicate that the mass flow and the pressure field in the tube (OA) are proportional to the fluid confinement ratio σ f = s f / S f . They both vanish if S f tends to infinity and, in this particular case, the tube (OA) can be replaced by an orifice opening into the free atmosphere, producing thus a pressure node at A. On the other hand, if S f tends to zero, the piping system is reduced to the single tube (BC). 2.3. Continuous systems 2.3.1
Modal added mass matrix
The coupled system [2.1] or [2.4], complemented by appropriate homogeneous boundary conditions concerning the structure and the fluid, can be discretized by using the natural modes of vibration of the structure in vacuum as a vector basis to describe the vibration of the structure coupled to the fluid. For the basic principles of the modal projection method the reader is referred to [AXI 05], Chapter 4. The displacement field of the structure loaded by the fluid is written as the modal series: ∞ X s ( r ; t ) = ∑ qi ( t )Φ i ( r )
[2.104]
i =1
Φi ( r ) designates the mode shapes in vacuum. Note carefully that here qi designate the generalized displacements of the structure, and not the volume velocities in a pipe.
The vibration equation of the structure is projected on the i-th mode shape to produce the following set of time-differential equations: ⌠
K s ( i, i ) qi + M s ( i, i ) qi = ⎮⎮
⌡(W )
p ( r0 ; t )Φ i ( r0 ) .n ( r0 ) dW
r0 ∈ (W )
[2.105]
On the other hand, the fluid pressure is governed by the partial differential equation: ∞ Δp = − ρ f ∑ qjΦ j ( r ).n ( r )δ ( r − r0 ) j =1
[2.106]
82
Fluid-structure interaction
By integrating [2.106] within a volume (Vf ) bounded by a closed surface and containing the current point r0 of the fluid-structure interface (W ), it follows that: ⌠ ⎮ ⎮ ⌡(Vf
)
⌠ div(grad p ) dV = ⎮⎮
⌡(W )
∞ grad p.n dW = − ρ f ∑ qjΦ j ( r0 ).n ( r0 )
[2.107]
j =1
As the relation [2.107] holds even if (Vf
)
vanishes, the equation [2.106] can be
expressed in terms of ordinary functions as the inhomogeneous system: Δp = 0 ∞ grad p( r0 , t ).n ( r0 ) = − ρ f ∑ qjΦ j ( r0 ).n ( r0 )
[2.108]
j =1
To solve the fluid problem, the system [2.108] must be complemented by the conditions to be fulfilled at the fluid boundaries other than (W ). Furthermore, due to linearity, the superposition principle holds and the solution can be expressed as a series of the type: ∞ p ( r ; t ) = ∑ b j (t ) p j ( r )
[2.109]
j =1
where p j is the solution of the particular problem: Δp j = 0 b j (t )grad p j ( r0 ).n ( r0 ) = − ρ f qjΦ j ( r0 ).n ( r0 )
[2.110]
+ (C.B.C)
Once more (C.B.C) stands for “Conservative Boundary Conditions” prescribed to the fluid volume. From [2.109] and [2.110] it is concluded that the space functions p j ( r ) may be used as a vector basis to describe the solution p ( r ; t ) , in the same way as the mode shapes of the structure are used to describe the displacement X s ( r ; t ) . However, in contrast with eigenmodes, the modulus of p j is not arbitrary since p j is the solution of a forced problem. The condition on the fluid-structure interface implies necessarily that: b j ( t ) = − ρ f q j ( t )
[2.111]
Substituting the series [2.109] into the structural equation [2.105] and with the aid of [2.111], we arrive at the following result: ∞
⌠
K s ( i, i ) qi + M s ( i, i ) qi = − ρ f ∑ qj ⎮⎮ j
⌡(W )
p j ( r0 ; t )Φ i ( r0 ) .n ( r0 ) dW
[2.112]
Inertial coupling
83
The right-hand side of [2.112] stands for the generalized force exerted by the fluid on the i-th vibration mode of the structure in vacuum. As the force is proportional to acceleration, it is inertial in nature. The proportionality coefficients are thus rightly interpreted as added mass coefficients given by: ⌠
M a ( i , j ) = ρ f ⎮⎮
⌡(W )
p j ( r0 ; t )Φi ( r0 ) .n ( r0 ) dW
[2.113]
The fact that M a ( i , j ) depends on the relative direction of the structural motion Φi and that of the normal n to the wall, expresses the directional aspect of the inertial coupling. Since the later is conservative in nature, the added mass matrix must be symmetrical, positive. Mathematical proof of these properties of primary importance can be outlined as follows. First, from the conditions [2.110] and [2.111] it results that: Φ i ( r0 ).n ( r0 ) = grad pi ( r0 ).n ( r0 ) [2.114]
which allows one to express [2.113] as: ⌠
M a ( i , j ) = ρ f ⎮⎮
⌡(W )
p j ( r0 ; t )grad pi ( r0 ).n ( r0 ) dW
[2.115]
On the other hand, the Laplacian operator complemented by conservative boundary conditions is self-adjoint, which leads to the Green identity (see Appendix A3): ⌠ ⎮ ⎮ ⌡(V )
⌠
p j Δpi dV − ⎮⎮
⌠ ⎮ ⎮ ⌡(W )
⌡(V )
pi Δp j dV = 0
⌠ p j grad pi .n dW - ⎮⎮
⌡(W )
pi grad p j .n dW = 0 ⇔ M a ( i, j ) = M a ( j , i )
[2.116]
Finally, since kinetic energy is a positive quantity, [ M a ] is necessarily positive or null:
[ q ] [ M a ][ q ] ≥ 0 T
∀ [ q ] ≠ [ 0]
[2.117]
As shown in [AXI 04], a necessary and sufficient condition for the inequality [2.117] to hold is that the eigenvalues of [ M a ] are all positive or null. Turning to the fluid-structure coupled problem [2.4], the natural modes of vibration of the structure coupled to the fluid are the nontrivial solutions of the matrix equation:
84
Fluid-structure interaction
2 ⎡ ⎡ ⎤⎤ ⎣ ⎡⎣ K s ( i, i )⎤⎦ − ωn ⎣ ⎡⎣ M s ( i , i ) ⎤⎦ + ⎡⎣ M a ( i, j ) ⎤⎦ ⎦ ⎦ [ qn ] = [0] φn = ⎡⎣Φ ⎤⎦ [ qn ]
[2.118]
The natural pulsation of the n-th mode is ω n . The associated mode shape is [ qn ] if expressed in the basis ⎡⎣Φ ⎤⎦ and φ n if expressed in the geometrical space of the structure. Φ n and φ n are either the same or not, depending whether the added mass matrix expressed in the ⎡⎣Φ ⎤⎦ basis is diagonal or not, that is depending whether the functional vectors p j and grad pi are orthogonal to each other, or not. On the other hand, since the generalized mass in fluid is larger than in vacuum, ω n is lower than the corresponding value Ω n in vacuum. Unfortunately, even if the mode shapes Φ n have simple analytical expressions, the fluid problem governed by the system [2.110] can be solved analytically only in a very few particular cases, as further illustrated in the remaining part of this chapter, which discusses a few problems involving various geometries, with the purpose of pointing out several interesting features of inertial coupling selected for their generic character. However, the presentation given here is by no means exhaustive, to keep it in a reasonable space. There exists an abundant literature specialized on the subject and the interested reader may be directed in particular to [BLE 79] to find a comprehensive and well documented compilation of added mass coefficients derived from either analytical or experimental studies. New information is also made available regularly in the technical literature, the Journal of Fluids and Structures in particular, indicating that the subject is not yet exhausted as a research field, even if numerical methods such as finite element and boundary element techniques are now becoming very popular in various fields of application. 2.3.2
Strip model of elongated fluid-structure systems
2.3.2.1 Cylindrical shells of revolution The geometric parameters and the displacement field of a cylindrical shell of revolution are defined in Figure 2.21, in agreement with the notations used in [AXI 05]. Here, the shell is filled with an incompressible liquid and the purpose is to analyse the natural modes of vibration of the shell. As a preliminary, it is recalled that in orthogonal curvilinear coordinates α , β , γ , the Laplacian of a scalar p is expressed as: Δp =
⎧⎪ ∂ ⎛ g β gγ ∂ p ⎞ ∂ 1 ⎨ ⎜ ⎟+ gα g β gγ ⎩⎪ ∂ α ⎝ gα ∂ α ⎠ ∂ β
⎛ gα g γ ∂ p ⎞ ∂ ⎜⎜ ⎟⎟ + ⎝ gβ ∂ β ⎠ ∂ γ
⎛ gα g β ∂ p ⎞ ⎪⎫ ⎜⎜ ⎟⎟ ⎬ ⎝ gγ ∂ γ ⎠ ⎭⎪
[2.119]
Inertial coupling
85
gα , gβ , gγ are the Lamé coefficients of the curvilinear metrics (see [AXI 05]) such
that the squared length of an infinitesimal line element is: ds 2 = ( gα ) dα + ( g β ) d β + ( gγ ) d γ 2
2
2
[2.120]
Figure 2.21. Circular cylindrical shell: geometry and global displacements
Adopting here cylindrical coordinates, the quadratic form [2.120] is readily found to be: ds 2 = dr 2 + r 2 dθ 2 + dz 2 ⇒ g r = 1 ; gθ = r ; g z = 1
[2.121]
Whence: 1 ⎪⎧ ∂ ⎛ ∂ p ⎞ ∂ ⎛ 1 ∂ p ⎞ ∂ ⎛ ∂ p ⎞ ⎪⎫ Δp = ⎨ ⎜ r ⎟+ ⎜ ⎟+ ⎜r ⎟⎬ r ⎪⎩ ∂ r ⎝ ∂ r ⎠ ∂ θ ⎝ r ∂ θ ⎠ ∂ z ⎝ ∂ z ⎠ ⎪⎭
[2.122]
A priori, the problem is three-dimensional in nature. However, the azimuthal or circumferential dimension can be removed by expanding the displacement and the pressure fields in Fourier series. Presentation of two-dimensional problems is postponed to subsection 2.3.4. Indeed, for mathematical convenience it is found appropriate to start with one-dimensional problems. This is made possible here by considering the case of a slender shell H/R >>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
Fluid-structure interaction
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
Fluid-structure interaction
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
Fluid-structure interaction
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
Fluid-structure interaction
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
Fluid-structure interaction
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
Fluid-structure interaction
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
Fluid-structure interaction
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
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Fluid-structure interaction
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
Fluid-structure interaction
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
Inertial coupling
103
Figure 2.31. Added mass ratio versus aspect ratio of the plate
As another exercise, the reader can also solve the 2D problem by expanding the plate displacement as the double Fourier series of the type: Z0 = β n ,m
2.3.4
n =+∞
m =+∞
n =1
m =1
⎛ nπ x ⎞ ⎛ mπ y ⎞ β n ,m sin ⎜ ⎟ sin ⎜ ⎟ ⎝ a ⎠ ⎝ b ⎠
∑ ∑
n m 4 ⎛ (1 − ( −1) )(1 − ( −1) ) ⎞ ⎟ = 2⎜ ⎟ π ⎜ nm ⎝ ⎠
[2.179]
Mode shapes modified by fluid inertia
In the preceding examples, the pressure field p j induced by the structural motion was suitably shaped to let the structural mode shapes Φi be unchanged. It is also of interest to study a few cases which give rise to a non-diagonal added mass matrix, leading thus to a modification of the “in vacuum” mode shapes. Generally, in such systems, the exact solution cannot be carried out analytically; this precisely because of the lack of orthogonality between p j and Φi : ⌠
M a ( i , j ) = ρ f ⎮⎮
⌡(W )
p j ( r0 ; t )Φi ( r0 ) .n ( r0 ) dW
≠ 0
[2.180]
M a ( i , j ) now differs from zero even if i and j are distinct, which prevents
successful application of the method of separation of variables. However,
104
Fluid-structure interaction
approximate solutions can be made available which contain most of the pertinent physical information, as illustrated in the following examples. 2.3.4.1 Rigid rod partly immersed in a liquid
Figure 2.32. Circular cylindrical rod partly immersed in water
The simplest example is that of a circular cylindrical rod supported by springs (stiffness coefficient K s ) and partly immersed in an infinite extent of liquid, as sketched in Figure 2.32. The rod is assumed to be homogeneous and rigid (length 2L, radius R and mass M s ). Vibration is supposed to be normal to the rod axis and in the plane of the figure. Denoting X the displacement of the centre of mass O of the rod and ψ the rocking angle of the rod about O, the modal equations of the system in vacuum are readily found to be: ⎡ ⎡2K s ⎢⎢ ⎣⎣ 0
0 ⎤ ⎡M −ω2 ⎢ s 2⎥ 2Ks L ⎦ ⎣ 0
0 ⎤ ⎤ ⎡ X ⎤ ⎡0 ⎤ = ⎥ I s ⎥⎦ ⎦ ⎢⎣ψ ⎥⎦ ⎢⎣0 ⎥⎦
[2.181]
where I s = M s L2 / 3 for a slender rod ( R / L << 1 ). The natural modes of vibration follow immediately as: 1. Mode of pure translation:
[Φ1 ]T = [1
0] ; Φ1 ( z ) = i
K1 = 2K s
; M1 = M s
; Ω1 =
2Ks Ms
[2.182]
Inertial coupling
105
2. Mode of pure rotation (rocking mode):
[Φ 2 ]T = [0 1] 2
K 2 = 2K s L
z ; Φ2 ( z) = i L
; M 2 = Is
6K s ; Ω2 = Ms
[2.183]
When the rod is immersed in water up to a depth H, it is necessary to account for the fluid inertia. Provided the system is slender enough (H/R>>1) an approximate solution can be easily derived by using the strip model, that is by adding the mass coefficient [2.144] to the structural mass per unit length of the rod. The kinetic energy of the fluid is thus approximated by: Eκ =
2 H −L 1 ρ f π R 2 ∫ ( X + zψ ) dz −L 2
[2.184]
The coefficients of the added-mass matrix follow as: M a (1,1) = ma H
; ma = ρ f π R 2
M a ( 2, 2 ) = 31 ma H(H 2 + 3L2 − 3LH)
[2.185]
M (1, 2 ) = M ( 2,1) = − ma H(2L − H) 1 2
Figure 2.33. Natural modes of vibration of the rod immersed at mid-height in a dense liquid
As expected, the added mass matrix is non-diagonal, except at full immersion of the rod. In agreement with the general formalism presented in subsection 2.3.1, the mode shapes [ϕ1 ] , [ϕ 2 ] in liquid are linear superposition of the mode shapes in vacuum, that is a compound of translation and rotation, as illustrated in Figure 2.33 which refers to the following numerical values: M s = 1 kg; K s = 10000 N/m; 2L = 1 m, H = L; ma H = 0.5 kg. In vacuum, the following modal quantities are found:
106
Fluid-structure interaction
f1 = 22.5 Hz ; M 1 = 1 kg ; f 2 = 39.0 Hz ; M 2 = 0.083 kg
[Φ1 ]T = [1 0] T ; [Φ 2 ] = [ 0 1]
In liquid up to mid-height they are transformed into: f1 = 18.0 Hz ; M 1 = 1.65 kg ; f 2 = 33.9 Hz ; M 2 = 0.12 kg ;
[ϕ1 ]T = [1 −0.47] [ϕ 2 ]T = [ 0.12 1]
The mode shapes in the physical space are thus: φ1 (z) = (1 − 0.47 z / L ) i and φ2 (z) = ( 0.12 + z / L ) i It is interesting to perform a parametric analysis of this system, as a function of the fluid height 0 ≤ H / 2 L ≤ 1 and of the mass-parameter μ a = ma /( M s / 2 L ) , which expresses the relative magnitudes of the distributed fluid and structural inertia. Figure 2.34 depicts the change of the modal frequencies of both modes (normalized by their corresponding values in vacuum), ω n / ω sn , as the fluid height increases, for several values of μ a . Of course, both modal frequencies decrease as H/2L increases, but not in a regular manner, depending on the mode and also on the magnitude of μ a . Notice that both modal frequencies decrease more dramatically, and at lower ranges of H/2L, as μ a increases, which is a mere consequence of the increasing magnitude of the fluid loading. On the other hand, for the second mode there is a plateau-range of H/2L values where no significant decrease of the modal frequency is displayed. Such behaviour is due to the low effectiveness of the fluid loading near the nodal region of this mode. As might be expected, this nodal region migrates towards the lower extremity of the bar as μ a increases.
Figure 2.34. Modal frequencies of the coupled system as a function of the fluid height and of the mass-parameter μ a
Inertial coupling
107
Figure 2.35. Modal masses of the coupled system as a function of the fluid height and of the mass-parameter μ a
Finally, it can be noticed that both reduced frequencies are similarly affected by the fluid when the full height of the bar is immersed. This can be easily understood by recalling that, when H / 2 L = 1 , the added mass matrix becomes diagonal and the mode shapes of the coupled system are the same as those in vacuum. All the previous comments may also be inferred from Figure 2.35, which depicts the increase of the modal masses of the fluid-coupled system M n / M sn , normalized by their corresponding values in vacuum, as a function of the fluid height and μ a . 2.3.4.2 Coaxial cylindrical shells of revolution
Figure 2.36. Experimental set-up for inertial coupling study of two coaxial shells
108
Fluid-structure interaction
The set-up sketched in Figure 2.36 was used by the first author [AXI 79] as an experimental validation of the FEM formulation implemented in CASTEM2000 [CAS 92] to deal with fluid-structure interaction problems. The FEM formulation will be outlined in Chapter 6, in relation to vibroacoustic coupling. Two thin cylindrical steel shells of revolution are welded coaxially on a thick (5 cm) circular steel plate. The interstitial space between the two shells is filled with water up to a height H which can be varied. The structure lies on a rubber sheet to produce isolation from the vibrations transmitted by the foundation of the test room. The external shell is excited by an electromagnetic shaker according to a sine signal slowly swept in frequency. The response of both shells is measured by using piezoelectric accelerometers glued on the shells along both a meridian and a parallel lines. The modes of lowest frequency correspond to the circumferential index n = 7 (concerning the natural modes of vibration of thin cylindrical shells, the reader is referred for instance to [AXI 05], Chapter 8).
Figure 2.37. Frequency plot of the shell modes n = 7, m = 1: experimental data
In vacuum, a pair of n = 7 modes were observed, namely an in-phase mode at frequency f1 41.5Hz and an out-of phase mode at f 2 44.5Hz . This indicates that the plate flexibility is not completely negligible even if the shells are apart from each other by 1cm only. Due to the coupling by the plate stiffness, the frequency of the out-of-phase mode is larger than that of the in-phase mode, as clarified latter in relation with Figure 2.40. The measured frequencies of these modes are plotted versus the interstitial water height in Figure 2.37. Figure 2.38 displays the fit of these data to numerical results produced by the FEM model. In both figures, the
Inertial coupling
109
frequencies are reported as abscissas and the water height as ordinates. As seen in Figure 2.39, where the computed radial displacement of both shells is plotted along a meridian, the axial out-of-phase mode shapes of the shells are deeply modified by the inertial coupling, and the in-phase mode shapes are not. Furthermore, a drastic change is observed to occur at about H c 200 mm concerning the relative phase of the vibration. Below H c , the frequency of the out-of-phase mode is higher than that of the in-phase mode but above H c the reverse occurs. This is due to the conflicting effect of the stiffness coupling by the plate and the inertial coupling by the fluid.
Figure 2.38. Frequency plot of the shell modes n = 7, m = 1: numerical results (FEM model) and experimental data
Near the inversion line (dot-dashed line in Figure 2.37) the two frequency branches come very close to each other, indicating that at H c the relative importance of the two coupling mechanisms is about the same. However, it is also important to emphasize that the two frequency branches of the coupled modes do not cross each other. A crossing would mean that the system becomes unstable, in full contradiction with the physics of the problem. This point can be clarified by considering a very simplified model using two degrees of freedom only, denoted q1 and q2 . The mode shapes in the radial direction are written as: U1n ( z, θ ) = q1ϕ ( z ) cos nθ
; U 2 n ( z, θ ) = q2ϕ ( z ) cos nθ
[2.186]
Accordingly, the added mass matrix is approximated by integrating the 2D matrix of the strip model [2.161] along the axial direction:
110
Fluid-structure interaction
[Ma ] =
ρ f π R 3 F ( H ) ⎡ 1 −1⎤ ⎢ −1 1 ⎥ n2h ⎣ ⎦
where F ( H ) =
H
⌠ ⎮ ⎮ ⌡0
ϕ
2
[2.187]
( z ) dz
In-phase mode: H = 0.600 m
Out-of-phase mode: H = 0.080 m Figure 2.39. Axial mode shapes at two water heights (abscissa: axial distance, ordinates radial displacement)
Inertial coupling
111
Out-of-phase mode: H = 0.300 m
Out-of-phase mode: H = 0.600 m Figure 2.39b. Axial mode shapes at two water heights (abscissa: axial distance, ordinates radial displacement)
As the model is not aimed to fit accurately experimental data but to capture the qualitative features of the coupling only, ϕ ( z ) can be approximated by the first mode shape of a clamped-free beam, or even by a simpler function, independently from the height of the water level. On the other hand, as an essential part of the
112
Fluid-structure interaction
model, the modes of each shell are coupled by a stiffness matrix which simulates the imperfect clamping of the plate. Physically, such a coupling is expected to increase the stiffness of the out-of-phase mode, in agreement with the deformations schematised in Figure 2.40. The left sketch (a) refers to the non deformed geometry. It can be understood qualitatively that an in-phase motion of the shells (sketch (b)), induces smaller strains in the plate than an out-of-phase motion, (sketch (c)). In principle, this could be modelled by a spring acting on the angle of deflection of the shells ψ 1 ,ψ 2 at the pinned ends which would induce the torque: K c (ψ 2 −ψ 1 )
[2.188]
Figure 2.40. Model of the imperfect clamping of the shells at the plate
A priori, a difficulty to implement such a model arises if the shell motion is described in terms of clamped-free modes, because ψ 1 ,ψ 2 would vanish. However, as the mechanical properties of the shells are very similar, the local effect of the plate flexibility is small and so is the frequency shift between the in-phase and outof-phase modes induced by the stiffness coupling. As a consequence, the coupling coefficient K c can be included directly in the model projected onto the clampedfree modes and its value adjusted to fit the frequencies measured in vacuum. In fluid, the modal problem to be solved is thus written as: ⎡ ⎡ K1 ⎢⎢ ⎣ ⎣ Kc
where:
Kc ⎤ ⎡M −ω2 ⎢ 1 ⎥ K2 ⎦ ⎣Mc
M c ⎤ ⎤ ⎡ q1 ⎤ ⎡0⎤ = ⎥ M 2 ⎦⎥ ⎦ ⎣⎢ q2 ⎦⎥ ⎢⎣0⎥⎦
[2.189]
Inertial coupling
113
K1 = K s1 ; K 2 = K s 2 Ma =
ρ f π R3F ( H ) n2h
; M 1 = M s1 + M a ; M 2 = M s 2 + M a ; M c = − M a
[2.190]
K s1 , K s 2 , M s1 , M s 2 are the generalized stiffness and mass coefficients of the mode n = 7 of the uncoupled shells in vacuum. As the [K] and [M] matrices are both positive definite, it is easily checked that the following inequalities hold:
K1 K 2 > 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
Fluid-structure interaction
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
Fluid-structure interaction
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
Fluid-structure interaction
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
Fluid-structure interaction
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
Fluid-structure interaction
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
Fluid-structure interaction
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
Fluid-structure interaction
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
Fluid-structure interaction
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 << M a initial acceleration is +2g upwards, where g designates the acceleration of Earth’s gravity and where the fluid friction is neglected. The last result is particularly appropriate to illustrate the basic distinction between the buoyancy force (Archimedes force) related to the so-called displaced mass concept M d , which is static in nature, and the inertia force related to the so-called added mass concept, which is dynamic in nature. 2.3.5.4 The immersed sphere used as an inverted pendulum
Figure 2.50. Spherical shell mounted as an inverted pendulum
Let us consider now a spherical shell which has a negative effective weight Pe , when immersed in water, due to the buoyancy force:
⎛ 4π R03 ρ f Pe = ⎜ M s − ⎜ 3 ⎝
⎞ ⎟⎟ g ⎠
and
4π R03 ρ f 3
> 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
Fluid-structure interaction
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
Fluid-structure interaction
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
Fluid-structure interaction
∂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
Fluid-structure interaction
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
Fluid-structure interaction
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]
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Fluid-structure interaction
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]
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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
Fluid-structure interaction
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.
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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)
If H is small in comparison with the wavelength, the dispersion equation [3.11] is drastically simplified, becoming: k2 =
ω2 ⇒ cψ = gH gH
(c
g
= cψ = c )
[3.34]
Thus, in shallow water the gravity waves are non-dispersive. The vertical profile of pressure is constant, to the first order of z / λ at least, and the displacement of the fluid particles is practically horizontal, as in the case of dilatation or sound waves. Actually, applying the thin fluid layer approximation introduced in Chapter 2, to the system [3.2], and assuming once more that the wave is uniform in the lateral direction, a one-dimensional equation is obtained which has the same form as that which governs the longitudinal waves in a straight beam of uniform cross-section, or the transverse waves along a tensioned string and, as shown later in Chapter 4, also that which governs the plane sound waves in a pipe of uniform cross-sectional area. Averaging of the 2D Laplacian over the fluid layer thickness gives:
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Fluid-structure interaction
H
⌠ ⎮ ⎮ ⎮ ⎮ ⌡0
⎛ ∂2 p ∂2 p ⎞ ∂ 2 p ∂p ⎜ 2 + 2 ⎟ dz = 0 ⇒ H 2 + ∂z ⎠ ∂x ∂z ⎝ ∂x
=0
[3.35]
z=H
Using then the free surface condition [1.69], repeated for convenience in [3.2], we obtain the following 1D wave equation in terms of pressure: H
∂ 2 p ∂p + ∂x 2 ∂z
=0⇒ z=H
∂2 p 1 ∂2 p − =0 ∂x 2 c 2 ∂t 2
[3.36]
where c = gH . Finally, using the free surface condition expressed as p = ρ f gZ 0 , the wave equation can be written in the two following equivalent forms: ∂ 2 Z0 1 ∂ 2 Z0 ∂ 2 Z0 ∂ 2 Z0 − = 0 ⇔ ρ gHS − ρ S =0 f f ∂x 2 c 2 ∂t 2 ∂x 2 ∂t 2
[3.37]
The last form is suitable for direct comparison with the wave equation which governs the longitudinal vibration of a straight beam of uniform cross-sectional area S: Es S
∂2 X ∂2 X − ρ S =0 s ∂x 2 ∂t 2
[3.38]
or the transverse vibration of a stretched string (tension force T0 ): T0
∂2Z ∂2Z − ρ S =0 s ∂x 2 ∂t 2
[3.39]
It may be worth mentioning briefly a few other consequences of the velocity law [3.12] and its “shallow water” version [3.34], which can be easily observed walking along the sea shore. The wind generated waves travelling from the open ocean toward a coast are generally not parallel to the coast when propagating on deep water. Nevertheless, as they approach a shelving beach, they are progressively refracted in such a way that the wave fronts are usually almost parallel to the shore by the time they break. Furthermore, the shape of the waves is by far not sinusoidal. In particular the leading edge is much steeper than the trailing edge, until it breaks into a foaming front, which is highly turbulent. Of course, such features are typical of finite amplitude waves, which cannot be modelled within the framework of the linear theory. Nevertheless, as pointed out later in subsection 3.2.6, the fact that in shallow water the wave speed increases with the water depth is also true in the nonlinear domain. Several other interesting features such as bores and hydraulic jumps can be explained qualitatively in a similar way, as discussed and nicely illustrated in the textbook by T.E. Faber [FAB 01], from which we borrowed the last example “which may be observed any day on a sandy beach, where a spent wave moves forward a foaming front over a receding layer of water left from the previous
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149
wave. Often one may see two such foaming fronts moving forward simultaneously, one behind the other, and the second front almost always catches up the first”. On the other hand, according to formula [3.34], the speed of such waves depends highly on H and then on the range of the wavelengths considered. As two extremes of practical interest, we may mention first the use of shallow water waves to simulate, in the laboratory, plane supersonic flows at low cost. Adopting for instance H = 0.1 m and f = 0.1 Hz, it follows λ ≅ 60 cm and c = 1 m / s , which is much smaller than the speed of sound in any elastic material, solid or fluid. At the opposite extreme, the oceanic waves known as tidal waves and tsunamis can be considered as shallow water waves whatever the actual sea depth may be. Indeed, the wavelengths of tide waves are within the range of 100 to 10000 km, while those related to tsunamis are within the range of 50 to a few hundred kilometres. 3.2.4
Application of the shallow wave theory to tsunamis
For a tsunami to occur it is necessary that either the sea floor, or the free surface, experiences a vertical shake over a sufficiently large area, whose dimensions are much larger than H. Such shakes can be generated by underwater earthquakes, landslides, volcanic bursts and even rapid changes in the atmospheric pressure, related typically to a travelling cyclonic area, see for instance [MAR 83] and [PEL 01]. Considering an oceanic basin of mean depth H = 4 km, which is typical in the Pacific ocean, the tsunami is found to propagate at a speed of 720 km/h, not much less than the cruising speed of the jet liners. Moreover, because the rate at which a wave loses its energy is inversely related to its wavelength (see Chapter 7), it turns out that tsunamis can also travel great transoceanic distances with limited energy loss. 3.2.4.1 Seismic tsunami waves As a first instructive exercise, we are interested here in describing the waves triggered by a sudden vertical motion of the sea floor based on the linear shallow wave model, restricted to the one-dimensional case for mathematical convenience. e Denoting by Z ( ) ( x; t ) the vertical displacement field of the sea floor and using the thin layer approximation already described in the subsection 3.2.1.3, equation [3.35] becomes: H
⌠ ⎮ ⎮ ⎮ ⎮ ⌡0
⎛ ∂2 p ∂2 p ⎞ ∂ 2 p ∂p ⎜ 2 + 2 ⎟ dz = 0 ⇒ H 2 + ∂z ⎠ ∂x ∂z ⎝ ∂x
= z=H
∂p ∂z
= −ρ f z =0
∂2Z ( ∂t 2
e)
[3.40]
which is readily transformed into: ∂2Z 1 ∂2Z 1 ∂2Z ( − 2 2 =− 2 2 ∂x c ∂t c ∂t 2
e)
[3.41]
150
Fluid-structure interaction
As a useful preliminary, let us consider first a concentrated quake of the type: Z ( ) = Z bδ ( x ) U ( t ) e
[3.42]
where U ( t ) designates the unit Heaviside step function. Solving the forced problem [3.41] is a straightforward task using the analytical tools already described in [AXI 04] and [AXI 05]. The equation [3.41] is transformed into: 2 ⎧ ∂ 2 Z ⎛ s ⎞ − Z =0 ⎪ ⎜ ⎟ 2 ∂x ∂ 2 Z ⎛ s ⎞ s ⎝c⎠ ⎪ δ − = − ⇔ Z Z x ⎨ b ( ) ⎜ ⎟ c2 ∂x 2 ⎝ c ⎠ s ∂Z ⎪ ∂Z − = − 2 Zb ⎪ ∂x x c ∂ ⎩ x =0+ x = 0− 2
[3.43]
Here s denotes the Laplace complex variable and the Laplace transformed quantities are identified by a superscripted tilde. Due to the symmetry about the origin, the system [3.43] can be split into two equivalent problems written as follows: ⎧ ∂ 2 Z + ⎛ s ⎞ 2 ⎪ 2 − ⎜ ⎟ Z + = 0 ⎝c⎠ ⎪ ∂x ⎨ s ⎪ ∂Z + = − 2 Zb ⎪ ∂x c 2 ⎩ x =0+
⎧ ∂ 2 Z − ⎛ s ⎞ 2 ⎪ 2 − ⎜ ⎟ Z − = 0 ⎝c⎠ ⎪ ∂x ⎨ s ⎪ ∂Z − = 2 Zb ⎪ ∂x c 2 ⎩ x = 0−
[3.44]
where Z + denotes the forward and Z − the backward wave. Solution to [3.44] is easily obtained as: = Z b e − sx / c ⇒ Z = Z b δ ( t − x / c ) Z + + 2c 2c Z Z / sx c b = Z e ⇒ Z− = b δ (t + x / c ) − 2c 2c
[3.45]
The particular waves [3.45] can be conveniently used to calculate those triggered by a source extended over a length 2L, much larger than H, as illustrated in the following example, where the source is assumed to be of the type: Z ( ) = Zb ( U ( x + L ) − U ( x − L )) U (t ) e
[3.46]
We may transpose here to space the informal reasoning presented in [AXI 04], Chapter 7, in the context of the transient excitations, to help the understanding of the convolution theorem. In the present context, the extended source can be viewed as a sequence of elementary concentrated sources of the type: dZ (
e)
( x − x0 ; t ) = Z b U ( t ) δ ( x − x0 )
− L ≤ x0 ≤ + L
[3.47]
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151
The elementary wave triggered by dZ ( ) is deduced from the result [3.45] as: e
dZ =
(
Zb δ ( t − ( x − x0 ) / c ) + δ ( t + ( x − x 0 ) / c ) 2c
)
[3.48]
The resulting wave is then obtained by summing the elementary waves. The result is the integral of convolution: Z ( x; t ) =
+∞
((
Z b ⌠⎮ ( U ( x0 + L ) − U ( x0 − L ) ) δ ( t − ( x − x 0 ) / c ) + δ ( t + ( x − x 0 ) / c ) 2c ⎮⌡−∞
) ) dx
0
[3.49] With the aid of the following formula, +∞
⌠ ⎮ δ ⎮ ⌡−∞
( ax + b ) dx =
+∞
1 ⌠⎮ 1 δ ( u ) du = ⎮ a ⌡−∞ a
[3.50]
we arrive at: Z ( x ; t ) = Z + ( x ; t ) + Z − ( x; t ) Zb ( U ( x + L − ct ) − U ( x − L − ct ) ) 2 Z Z − ( x; t ) = b ( U ( x + L + ct ) − U ( x − L + ct ) ) 2 Z + ( x; t ) =
[3.51]
Figure 3.3 shows the resulting wave at a few distinct times, for the numerical values specified in the figure. As expected, the outgoing and incoming waves separate from each other from time ts = L / c 200 s . An especially vicious aspect of such “killer waves” is that in deep water their height is generally in the range of a few tens of centimetres, so very difficult to detect; in particular a boat can ride over the worst tsunami without even noticing them. Then, as they approach the shallow water near a coast the radiated power or energy flux remains almost unchanged while the wave speed and the wavelength decreases, hence the wave height increases. To become aware of the importance of this treacherous effect, let us consider a tsunami of height Z 0 = 60 cm in deep water H 0 = 4000 m. Starting from the relation [3.33] and making the reasonable assumption that the radiated power is essentially constant during propagation, the height Z1 of the wave for a water depth H1 is given by the following relation: 1/ 4
⎛H ⎞ Z1 = Z 0 ⎜ 0 ⎟ ⎝ H1 ⎠
[3.52]
152
Fluid-structure interaction
Figure 3.3. Resulting wave profile at a few distinct times corresponding to ct = 0, L/2, L and 3L/2 respectively
Accordingly, a wave of about four meters height can be expected at a mean water depth of two meters. As the wavelengths of tsunami waves are also much larger than those of deep water waves, when it reaches the coast a tsunami wave often appear as a rapidly rising or falling tide. Clearly, the linear one-dimensional model used here is totally insufficient to describe the large variety of behaviours which are observed in real tsunamis. In particular, the transformation of the wave when approaching the coast can vary greatly, depending on the floor profile and the shape of the bay. As a consequence, the height of a tsunami wave can vary greatly along a coast, reaching in some cases more that ten meter and near the coast a tsunami can also appear as series of breaking waves, instead of as a tide wave. Nevertheless, the simplistic linear equation [3.41] is still sufficient to capture the basic features which explain the names given to this kind of waves: the Japanese name “tsunami” means “harbour wave”, referring thus directly to the shoaling effect according to which a tsunami wave remains undetected beyond the continental shelf and appears as an impressive water wall, or as series of large and even huge waves when approaching the shoreline. Tsunamis are also often called tidal waves. Though the name is irrelevant so far as the driving mechanism is concerned - true tidal waves result from the imbalanced gravitational forces exerted by the moon, sun and planets - tsunami waves also invade the beach and flood low lying land over a large distance from the coast, and then recede a few ten minutes later far beyond the normal mean sea level. 3.2.4.2 Meteorological tsunamis In many cases, the tsunami sources move with variable speed and direction. This is typically the case for meteorological tsunamis (see [PEL 01] from which the present considerations are largely inspired). As shown below, the wave equation
Surface waves
153
forced by a fluctuating pressure prescribed at the free level is of the same type as equation [3.41]. However, the response presents new interesting features with respect to the non moving source case, which are worthwhile to describe. Starting from the unperturbed state of equilibrium where the liquid layer has a uniform depth H, and the atmospheric pressure has the uniform and static value P0 , the atmospheric perturbation is characterized by the pressure field: ⌠
Pa ( x; t ) − P0 = ⎮⎮
+∞
⎮ ⌡−∞
P(
e)
(ξ ) δ (ξ − ( x − Vt ) ) dξ = P ( e) ( x − Vt )
[3.53]
where V is the source speed in the Ox direction, assumed to be constant for analytical convenience. The Dirac distribution is used to concentrate the excitation at the running coordinate x = Vt, as already explained in [AXI 04], Chapter 4, in the context of moving loads on a flexible bridge, which present some striking similarities with the present problem, as shown below. As a special case, a static perturbation corresponds to:
(
lim P (
e)
( x − Vt ) ) = Pa ( x )
[3.54]
V →0
The free surface response to Pa ( x ) is a static elevation of the free surface with respect to H, given by: Za = −
Pa ρf g
[3.55]
The relation [3.55], known as the law of inverse barometer, can be used to trace the “static” atmospheric perturbations based on the data concerning changes in the sea level obtained from space. On the other hand, the condition at the perturbed free surface becomes: ⎛ ∂ p⎞ e = P ( ) ⎜p + g ⎟ ∂ z ⎝ ⎠ z = H + Za
[3.56]
Using once more the thin layer model, we arrive at the forced wave equation: ∂2 p 1 ∂2 p 1 ∂2 P( − = − ∂x 2 c 2 ∂t 2 c 2 ∂t 2
e)
[3.57]
The quasi-static response of the free surface to P ( by the static equilibrium condition P
(e)
e)
is an elevation Z ( (e)
( x; t ) = − ρ f gZ ( x; t ) .
e)
( x; t )
given
Recalling that
p = ρ f gZ , where Z is the elevation of the wave with respect to the perturbed
154
Fluid-structure interaction
reference level H + Z a , the equation [3.57] takes on the same form as [3.41] with a sign change in the source term: ∂2Z 1 ∂2Z 1 ∂2Z ( − = + ∂x 2 c 2 ∂t 2 c 2 ∂t 2
e)
[3.58]
For the sake of simplicity, let us consider a source shaped as a rectangular pulse, moving at constant speed V in the Ox direction. The source is thus expressed analytically as: ()= Z ( ) = Z 0 ( U ( x − Vt + L ) − U ( x − Vt − L ) ) ⇒ Z e
e
s( x−L) s( x + L) ⎞ − Z0 ⎛ − V V − e e ⎜⎜ ⎟⎟ s ⎝ ⎠
[3.59]
The wave equation [3.58] forced by [3.59] and transformed by Laplace is: 2 s( x + L) ⎛ − s ( xV− L ) ⎞ − ∂ 2 Z ⎛ s ⎞ s V − = − Z Z e e ⎜ ⎟⎟ 0 ⎜ ⎟ 2 2 ⎜ ∂x c ⎝c⎠ ⎝ ⎠
[3.60]
where Z denotes the Laplace transform of Z. It is recalled that to solve the linear ordinary differential equation [3.60], the general method consists in adding a particular solution of the forced equation to the general solution of the homogeneous equation. So the solution is found to be a linear superposition of a forced wave, denoted Z F , and of two free waves, denoted Z + and Z − . Furthermore, the integration constants present in the free waves are adjusted to fit the initial conditions. The form of the source term encourages us to try a forced solution of the kind: s( x + L) ⎛ − s( x − L) ⎞ − = αZ ⎜e V − e V ⎟ Z F 0 ⎜ ⎟ ⎝ ⎠
[3.61]
Substituting [3.61] into [3.60] allows us to determine the constant α. The image of the forced wave is found to be: = Z F
s( x − L) s( x + L) ⎞ − V 2 Z0 ⎛ − V V − e e ⎜ ⎟⎟ 2 2 ⎜ c −V s ⎝ ⎠
[3.62]
whence in the time domain: Z F ( x; t ) =
V 2 Z 0 ( U ( x + L − Vt ) − U ( x − L − Vt ) ) c2 − V 2
V 2 Z ( ) ( x − Vt ) c2 − V 2 e
=
[3.63]
If V tends to zero, the forced wave vanishes, confirming if necessary that Z F refers to the perturbed free surface at H + Z 0 . However in [PEL 01], the quasi-static
Surface waves
response Z (
e)
( x − Vt ) is
155
added to the forced response [3.63], giving the following
elevation profile: ⎛ V2 ⎞ e c2 e Z F ( x; t ) = ⎜ 2 + 1⎟ Z ( ) ( x − Vt ) = 2 Z ( ) ( x − Vt ) 2 2 c −V ⎝ c −V ⎠
[3.64]
If V tends to zero, Z F tends to the static response of the free surface to the static perturbation Z (
e)
( x ) = Z a , so
Z F stands for the total elevation of the free surface,
reckoned from the non perturbed position H. Turning now to the free waves, the general solution of the homogeneous equation [3.60] is: +Z = A ( s )e − sx / c + A ( s )e + sx / c ⇒ Z + Z = A ( x − ct ) + A ( x + ct ) Z + − + − + − + −
[3.65]
The complete solution Z = Z F + Z + + Z − which complies with the initial conditions: Z ( x;0 ) = 0;
∂Z ∂t
=0
[3.66]
t =0
is finally obtained as: Z ( x; t ) =
e e e c 2 Z ( ) ( x − Vt ) c ⎛ Z ( ) ( x − ct ) Z ( ) ( x + ct ) ⎞ + − ⎜ ⎟ ⎟ c2 − V 2 c +V 2 ⎜⎝ V − c ⎠
[3.67]
The forced wave travels at the same speed as the source, while the free waves are travelling at the shallow water wave speed, forward and backward from the source. One remarkable result is the infinite amplitude of the forced wave predicted by the linear theory at the coincident speed V = c, also called resonant speed in reference to the classical oscillatory resonances. Actually, it is appropriate to refer here to the problem of a beam loaded by a travelling force mentioned just above. It is recalled that for a “pinned-pinned” beam of length L, the following infinite sequence of modal resonance conditions was found to be: Ωn =
nπ V = ω n ; ∀n ≥ 1 L
[3.68]
were ω n stands for the natural pulsations of the beam modes. Though the problem was treated by considering the case of bending modes, the results can be immediately transposed to the case of a tensioned string, which is more relevant to the present problem. As a consequence, the condition of resonance [3.68] can be further expressed as: nπV nπ c V = ⇔ =1 L L c
[3.69]
156
Fluid-structure interaction
Here c = T0 / ρ s S stands for the speed of the string waves, see equation [3.39]. On the other hand, the question as to whether such a result also holds in the framework of a nonlinear model, or not, naturally arises. This point is the object of the next subsection. 3.2.4.3 Nonlinear limitation of resonant waves A simplified nonlinear model adopted in [PEL 01] consists in averaging the condition of fluid incompressibility over the deformed layer thickness, which yields: H +Z
⌠ ⎮ ⎮ ⎮ ⎮ ⌡0
∂u ⎛ ∂u ∂w ⎞ + ⎜ ⎟ dz = ( H + Z ) + w z = H + Z = 0 ∂x ⎝ ∂x ∂z ⎠
[3.70]
where u denotes the mean fluid velocity in the horizontal Ox direction, as averaged over the depth of the fluid layer and w that in the vertical direction. Using the nonlinear free surface condition [1.71], it follows that: w z=H +Z =
D Z ∂Z ∂Z = +u Dt ∂t ∂x
[3.71]
On the other hand, based on the order of magnitudes [1.97], w is smaller than u by a ratio L/H so the w terms are neglected in the longitudinal momentum equation, which is thus written as: ∂u ∂u 1 ∂P +u + =0 ∂t ∂x ρ f ∂x
[3.72]
The vertical momentum equation is replaced by the hydrostatic equilibrium equation. Thus the pressure P is given by the barometric law: P = Pa + ρ f g ( Z − z )
[3.73]
Substituting [3.71] and [3.73] into [3.70] and [3.72] respectively, the following nonlinear system of equations is obtained: ∂Z ∂ ( H + Z ) u + =0 ∂t ∂x ∂u ∂u ∂Z 1 ∂Pa +u + g =− ∂t ∂x ∂x ρ f ∂x
[3.74]
Assuming a solution function of the running coordinate r = x − Vt , the system [3.74] is conveniently transformed into the following nonlinear algebraic system:
Surface waves
157
( H + Z ) u − VZ = 0 gZ +
u2 P − Vu = − a 2 ρf
[3.75]
By eliminating u between the two equations [3.75] and after a few manipulations, the following dimensionless algebraic equation is arrived at: 2ς 3 + ( 2 P + 4 − F ) ς 2 + ( 4 p + 2 − 2F ) ς + 2 P = 0
[3.76]
where F = V 2 / gH = (V / c ) is a Froude number, which can also be understood as 2
a Mach number squared, since c is the speed of the shallow water waves. The reduced magnitude of the depression is P = Pa / ρ f gH and ς = Z / H is the reduced elevation of the free surface. On the other hand, if the same mathematical procedure is applied to the linear model, the following linear equation is obtained: ς (1 − F ) + P = 0 ⇒ ς =
P (1 − F
[3.77]
)
Fixing P and considering F as a free parameter, the root of equation [3.77] can be viewed as a transfer function, which gives the wave elevation as a function of the Froude number. In the linear case, a peak of resonance of unlimited amplitude is observed at the resonance, in conjunction with an abrupt change of phase, just as in the case of the transfer function of an undamped harmonic oscillator, see Figure 3.4a. In a similar way, the roots of the cubic equation [3.76] define three branches. It turns out that only one of them is physically meaningful and thus plotted in Figure 3.4b.
(a)
(b)
Figure 3.4. Resonance curve (H = 92 m, c = 30 m/s), (a) linear model, (b) nonlinear model. Z / Z a is the reduced wave height and Fr means the Froude number
158
Fluid-structure interaction
The remarkable result is found that the nonlinearity limits the free surface elevation at resonance. As indicated in Figures 3.4, the wave height is scaled by the static water elevation induced by the depression Z a = P / ρ f g 50 cm and the dynamic water elevation can be about ten times the static value, see Figure 3.4b. 3.2.5
Deep 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
Fluid-structure interaction
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
[3.91]
Therefore the contribution arising from the k0 vicinity is expressed as: F ( k0 ) = f ( k0 ) e k1 = k0 − Δk ;
⌠
k2
iψ ( k0 ) ⎮
⎮ ⎮ ⌡k1
e
i
( k − k0 )2 2
k 2 = k 0 + Δk
ψ ′′( k0 )
dk
[3.92]
Δk << k0
At this step, it is useful to introduce the following change of variable which will allow one to express the integral in terms of known functions, as shown below: κ2 = −
( k − k0 )
ψ ′′ ( k0 ) 2 2
Substituting [3.93] into [3.92] yields:
[3.93]
162
Fluid-structure interaction
F ( k0 ) = f ( k0 ) e
iψ ( k0 )
F ( k0 ) = f ( k0 ) e
iψ ( k0 )
−2 ⌠ κ 2 − iκ 2 ⎮ e dκ ψ ′′ ( k0 ) ⎮⌡κ1
[3.94]
κ2 κ ⌠ 2 ⎞ −2 ⎛ ⌠⎮ 2 ⎮ ⎜ ⎮ cos (κ ) dκ − i ⎮ sin (κ 2 ) dκ ⎟ ⎮ ⎟ ψ ′′ ( k0 ) ⎜⎝ ⎮⌡κ1 ⌡κ1 ⎠
where: κ 1 = ( k1 − k0 )
−ψ ′′ ( k0 ) −ψ ′′ ( k0 ) ; κ 2 = ( k 2 − k0 ) 2 2
[3.95]
The integrals present in [3.95] can be expressed in terms of the Fresnel integrals defined as: y
C ( y) =
⌠ ⎮ ⎮ ⎮ ⎮ ⌡0
⌠
y
⎮ ⎛ π u2 ⎞ ⎛ π u2 ⎞ ⎮ cos ⎜ ⎟ du; S ( y ) = ⎮ sin ⎜ ⎟ du ⎝ 2 ⎠ ⎝ 2 ⎠ ⎮
[3.96]
⌡0
Using the change of variable: κ2 =
π u2 2
[3.97]
the integral [3.94] reads as: F ( y2 , y1 ) = f ( k0 ) e
iψ ( k0 )
−π C ( y2 ) − C ( y1 ) − i ( S ( y2 ) − S ( y1 ) ) ψ ′′ ( k0 )
(
)
[3.98]
Furthermore, the Fresnel integrals have the following asymptotic values: C ( +∞ ) = S ( +∞ ) =
1 2
; C ( −∞ ) = S ( −∞ ) = −
1 2
[3.99]
thus letting y2 tend to + ∞ and y1 tend to −∞ , which means that the whole range of real wave numbers from −∞ to + ∞ is accounted for, we arrive at: F ( +∞, −∞ ) = f ( k0 ) e
iψ ( k0 )
⎛
π⎞
i ⎜ψ ( k0 ) − ⎟ −π −2π 4⎠ (1 − i ) = f ( k0 ) e ⎝ ′′ ψ ( k0 ) ψ ′′ ( k0 )
[3.100]
Application to the deep water waves is straightforward. The phase function is: ψ ( k ) = t kg − kx
The condition of stationary phase yields the following intermediate results:
[3.101]
Surface waves
dψ dk
= k = k0
t g gt 2 − x = 0 ⇒ k0 = 2 2 k0 4x
2
t g −3/ 2 dψ d 2ψ = − ⇒ k 4 dk 2 dk 2
k = k0
; ψ ( k0 ) =
gt 2 4x
163
[3.102]
2 x3 =− 2 gt
Let us consider now the outgoing wave triggered by an initial displacement field, shaped as the rectangular pulse: Z ( x;0 ) = Z 0
Δ x0 Δx
Δx ⎞ Δx ⎞⎫ ⎧ ⎛ ⎛ ⎨U ⎜ x + ⎟−U⎜x− ⎟⎬ 2 2 ⎠⎭ ⎝ ⎠ ⎝ ⎩
[3.103]
which is scaled in such a way that if the width Δ x tends to zero, Z ( x;0 ) tends to the Dirac pulse: lim Z ( x;0 ) = Z 0 Δ x0δ ( x )
[3.104]
Δ x →0
The Fourier transform of the rectangular pulse is: Δx Z ( k ;0 ) = Z 0 0 Δx
+ Δx / 2
⌠ ⎮ ⎮ ⌡− Δ x / 2
e − ikx dx = 2 Z 0 Δ x0
sin ( k Δ x / 2 ) kΔx
[3.105]
Thus, the initial displacement field can be written in terms of the spectral (or wave number) components by using the inverse Fourier transform: 2Z Δ x Z ( x;0 ) = 0 0 π
+∞
⌠ ⎮ ⎮ ⎮ ⎮ ⌡−∞
sin ( k Δ x / 2 ) + ikx Z Δx e dk = 0 0 kΔx π
+∞
⌠ ⎮ ⎮ ⎮ ⎮ ⌡0
sin ( k Δ x / 2 ) cos kx dk [3.106] kΔx
Identifying the wave profile at time t = 0 with the halved initial displacement field, we can determine α k : Z ( x;0 ) = αk =
+∞
⌠ ⎮ ⎮ ⌡0
Z Δx α k cos kx dk = 0 0 π
Z 0 Δ x0 sin ( k Δ x / 2 ) π kΔx
+∞
⌠ ⎮ ⎮ ⎮ ⎮ ⌡0
sin ( k Δ x / 2 ) cos kx dk ⇒ kΔx
[3.107]
With the aid of the intermediate results [3.100] [3.102] and [3.107], the complex amplitude of the pressure field is found to be of the following asymptotic form: ⎛ gt 2 π ⎞
ρ gZ Δ x sin ( k0 Δ x / 2 ) ⎛ t g ⎞ i ⎜⎜⎝ 4 x − 4 ⎟⎟⎠ p ( x; t ) = f 0 0 ⎜⎜ ⎟⎟ e π k0 Δ x ⎝x x⎠
[3.108]
164
Fluid-structure interaction
Retaining the real part of [3.108] only, the real profile of the pressure is finally obtained as: p ( x; t ) =
ρ f gZ 0 Δ x0 ⎛ sin k0 Δ x / 2 ⎞ ⎛ t g ⎞ ⎛ ⎛ gt 2 ⎞ ⎛ gt 2 ⎞ ⎞ + cos sin ⎜ ⎟ ⎜ ⎜ ⎟⎜ ⎜ ⎟ ⎜ ⎟⎟ ⎟ π ⎝ 4x ⎠ ⎝ 4x ⎠ ⎠ ⎝ k0 Δ x ⎠ ⎝ x 2 x ⎠ ⎝
[3.109]
Letting Δ x tend to zero and dividing [3.109] by the scaling factor 2 ρ f gZ 0 Δ x0 , we obtain the profile of the free surface induced by prescribing at time t0 = 0 a displacement of unit magnitude concentrated at x0 = 0 . More generally, if x0 and t0 differ from zero, the following Green function is obtained: ⎛ ⎛ g ( t − t0 ) 2 ⎞ ⎛ g ( t − t0 )2 ⎞ ⎞ ⎛ t − t0 ⎞ g ⎜ G ( x , x0 ; t , t 0 ) = ⎜ cos + sin ⎜ ⎟ ⎜ ⎟⎟ ⎟ ⎜ 4 ( x − x0 ) ⎟ ⎜ 4 ( x − x0 ) ⎟ ⎟ ⎝ x − x0 ⎠ 2π ( x − x0 ) ⎜⎝ ⎝ ⎠ ⎝ ⎠⎠
[3.110]
Actually, the physical dimension of G is not a length, but the reciprocal of a length, because the source is a Dirac δ ( x − x0 ) .
Figure 3.6. Space profile of the gravity wave on deep water canal
Figure 3.6 shows a space profile of a deep water wave in a straight canal. The wave is triggered by an initial displacement Z 0 = 5 cm applied as a rectangular pulse Δ x0 = 10 cm and the profile is plotted five seconds later. The dimensionless distance from the source is defined as ξ = 2x/gt 2 .
Surface waves
165
Figure 3.7. Time profile of the surface wave
Figure 3.7 is a plot of the time profile at a fixed point along the canal for the same case. Dimensionless time is defined as τ = t 2x/g . From these figures the following points of interest can be noticed: 1.- As we approach the source the smaller the wavelength the higher the water crests. In reality, the wave magnitude is limited because the source cannot radiate wavelengths much shorter than the source width Δ x0 . Accordingly, instead of the asymptotic results [3.109] or [3.110], it would be necessary to perform the integration in a finite range −kc , + kc of wave numbers, by using the more exact formula [3.98] than its asymptotic form [3.100]. 2
2.- As t increases, the wave spreads out in proportion to t and its magnitude decreases in such a way that the area ∫ Zdx remains unchanged. This because the model is conservative. 3.- At a given location, the undulations follow each others as time elapses at an increasing frequency and with growth in amplitude. Again, according to a more realistic model, which would account for a cut-off value kc of the wave number, one would observe, far from the source wave, packets travelling at the group velocity: cg ∞ =
1 g / kc 2
[3.111]
166
Fluid-structure interaction
3.2.5.2 Wake of a moving boat and Kelvin wedge Another intriguing peculiarity of the deep water waves, worth to be briefly mentioned here, is the so-called Kelvin wedge related to the wake created by a boat moving in open water. The detailed wave pattern left behind any moving boat results from an intricate superposition of distinct waves excited at the bow and at the stern, eventually also at other parts of the hull. Nevertheless, it turns out that the wake as a whole is always confined within a wedge whose apex is at the ship’s bow and the opening angle is independent of the boat speed, in contrast with the acoustic shock front cone created by a projectile moving at supersonic speed, or that associated with the Cerenkov light excited by a charged particle travelling in a material at a greater speed than the light. Both the Mach and the Cerenkov angles are related to the wave speed c and that of the projectile denoted V, by the formula: α = sin −1 ( cψ / V )
[3.112]
As will be justified a little later, the law [3.112] holds whenever the waves are non dispersive. Turning back to the case of the deep water waves, let us consider those excited at the bow of a ship travelling at constant velocity V. A priori, a continuous wave spectrum can be excited and the related wave vector k can point to any direction in the plane of the free surface. However, to obtain constructive interferences between such elementary waves some restrictive conditions have to be fulfilled. First, to be reinforced, a wave emitted in a direction whose inclination to the travelling direction of the boat is π / 2 − α , and hereafter referred to as a “α-wave”, must comply with the condition: cψ (α ) = V sin α
[3.113]
in such a manner that the elementary wave emitted at (O) and time t=0 is in-phase with that emitted earlier at (O’) and time t = −τ , as shown schematically in Figure 3.8, where planes of constant phase related to the “α-wave” successively emitted at times −2τ , − τ , 0 by the travelling source are represented. Reinforcement of the individual waves occurs if the condition P ′′P ′ / cψ (α ) = O ′′O' / V ⇔ cψ (α ) = V sin α is fulfilled. With the aid of the dispersion equation [3.78], the “α-wave” number is found to be: kα =
g V sin 2 α 2
[3.114]
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167
Figure 3.8. “α-wave” system excited at the bow
If cψ is constant, the condition [3.113] leads naturally to the law [3.112]. Of course, this is not the case if cψ varies as wave reinforcement at a given point can originate from various wave components travelling at different speeds and angles. Considering a point B, making an angle β with the travelling direction and at distance r from the current position (O) of the bow, the phase angle of B associated with the “α-wave” is: ψ = kα .r = kα r sin (α − β ) [3.115] The condition of constructive interference reads now as a stationary phase condition. With the aid of [3.114] and [3.115] by performing a few elementary manipulations, it can be expressed as the following explicit function: dψ ⎛ tan α ⎞ = 0 ⇒ β = tan −1 ⎜ ⎟ 2 dα ⎝ 2 + tan α ⎠
[3.116]
which is plotted in Figure 3.9 in the interval 0°, 90°. The curve has a single extremum at α 0 = 54.7° ; β 0 = 19.5° , which means that the elementary waves outside the range −19.5° ≤ β ≤ 19.5° defining the Kelvin wedge cancel and this independently from the boat speed.
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Fluid-structure interaction
Figure 3.9. Stationary phase function for the “α-wave” system at a point P (β, r)
3.2.6
Water waves at intermediate depths: solitary waves
At intermediate depths, the most remarkable feature certainly is the occurrence of nonlinear waves known as solitary waves. Though this large and complex subject lies largely beyond the scope of the present book, it would be frustrating to omit it completely, at least because of its interest to various distinct fields of physics, where they are most often referred to as solitons. Solitary wave phenomenon was discovered in 1834 by Scott Russel, an engineer who was in charge of designing barges for use in the Union canal near Edinburgh. As a barge rapidly drawn by a pair of horses stopped abruptly, he noticed that water surged ahead in the form of a single wave whose height shape and speed, of about 8 to 9 miles/hour, remained virtually unchanged for more than a mile. In 1895, Korteweg and de Vries [KOR 95] derived the following nonlinear equation able to explain the existence of such solitary waves: ⎛ ∂Z H 2 ∂ 3 Z ∂Z 3 ∂Z ⎞ ± c⎜ + + Z ⎟=0 3 6 ∂x 2 H ∂x ⎠ ∂t ⎝ ∂x
[3.117]
where c = gH stands for the shallow water wave speed and where the third and fourth terms on the left-hand side of equation [3.117] are small (but not negligible) terms assumed to be of the same order of magnitude, that is: ς=
Z0 H2 2 = η2 < 1 H λ
[3.118]
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169
Of course, the parameter η has the same meaning as the dimensionless depth kH. However, it must be emphasized that to establish the Korteweg and de Vries equation (in short KdV) is by no means a trivial task. It requires several approximations and involves a perturbation expansion in the two physical parameters ς and η . The interested reader can be referred for instance to [JOS 02] for a presentation of the procedure. Nevertheless, it is rewarding to discuss briefly the different terms in the Korteweg and de Vries equation [3.117]. First, in the simplest linear form, ∂Z ∂Z ±c =0 ∂t ∂x
[3.119]
it is not difficult to recognize a pair of first-order wave equations as they express the same relation between the time and the space derivatives as that expected for a linear and non dispersive wave. Indeed, [3.119] is verified by any forward wave of the type Z + = f ( t − x / c ) and by any backward wave of the type Z − = g ( t + x / c ) , as can be easily checked by substitution into the second-order wave equation [3.41]. Then, considering the improved linear equation, ⎛ ∂Z H 2 ∂ 3 Z ⎞ ∂Z ± c⎜ + ⎟=0 6 ∂x 3 ⎠ ∂t ⎝ ∂x
[3.120]
it is easily verified that the last term induces dispersion. Substituting a harmonic and progressive solution into [3.120], the following dispersion relation is obtained: 2⎞ ⎛ 1 ω = ± kc ⎜ 1 − ( kH ) ⎟ ⎝ 6 ⎠
[3.121]
where again, the appropriate sign depends on the travelling direction of the wave. Incidentally, it may be noticed that we can arrive at the same result as [3.121] by expanding [3.11] to a second-order in powers of kH. Finally, to analyse the physical effect related to the nonlinear term of the KdV equation [3.117], the latter is simplified into: ∂Z ∂Z 3c ± (c + β Z ) = 0 where β = ∂t ∂x 2H
[3.122]
It turns out that solving [3.122] is rather easy, once an educated guess is made. Considering for instance the forward wave, as β tends to zero the solution must tend to Z + = f ( t − x / c ) . Hence it is tempting to try a nonlinear solution of the type: ⎛ x ⎞ Z+ = f ⎜ t − ⎟ ⎝ c+ βZ ⎠
[3.123]
170
Fluid-structure interaction
Figure 3.10a. Superposed linear profiles of the sine surface wave
Figure 3.10b. Superposed nonlinear profiles of the sine surface wave
which is the good guess, as direct substitution into [3.122] shows. From the physical standpoint, the solution [3.123] means that the phase velocity increases with the wave amplitude, which is in qualitative agreement with the linear result [3.12], according to which phase velocity of the linear wave increases with the water depth.
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171
Going a step further in the analysis, it is important to emphasize that even if β Z is much smaller than c, its effect is cumulative. Indeed, the crest of the wave travels faster than the trough, hence the crest tends to catch up with the trough, as in the afore-mentioned example of the foaming fronts of spent waves on a sandy beach (see subsection 3.2.3). In consequence, the wave front steepens, as illustrated in Figures 3.10a,b. The linear wave shown in Figure 3.10a is a sine wave Z 0 sin ω ( t − x / c ) plotted at different times versus x, the numerical values are H = 10 m, λ = 100 m and Z 0 = 0.5 m , hence c = 100 m/s and f = 0.1 Hz. To help visualization, a suitable constant has been added to each individual plot and the dotted lines stand for the true zero level of each individual wave profile. The dashed lines represent the time-evolution of the wave crests. The corresponding nonlinear profiles are computed using [3.123] and plotted in Figure 3.10b. As is conspicuous in such plots, the effect of nonlinearity is to distort the sine wave, in such a manner that the fronts are steeper and the trailing edge softer than in a sine profile. Actually, equation [3.123] can have several positive roots in a certain range of parameters, at least. However multivalued waveforms are physically meaningless. Very steep wavefronts can be interpreted in terms of shock waves. Moreover, they lead to large viscous dissipation, which invalidates the present model. Considering now the full KdV equation [3.117] and assuming once more that it has a solution of the general type Z + = f ( t − x / c ) , it can be proved that such a solution is (see for instance [JOS 02], [SCO 03]): ⎛ x ± cψ t ⎞ Z = Z 0sech 2 ⎜ ⎟ ⎝ L ⎠ ⎛ H3 ⎞ Z ⎞ ⎛ where cψ = c ⎜ 1 + 0 ⎟ and L = 2 ⎜ ⎟ ⎝ 2H ⎠ ⎝ 3Z 0 ⎠
[3.124]
Such a solution is called a solitary wave. It is quite remarkable that it can travel very long distances still preserving its shape. Due to the condition [3.118], the phase speed is slightly faster than that of the corresponding linear shallow water wave. On the other hand, the ability of the solitary wave to preserve its shape during propagation can be understood qualitatively as resulting from the contradicting effects of dispersion and “wave steepening”, which in some circumstances can be exactly balanced, to give rise to a solitary wave. Figure 3.11 illustrates the space profile of the corresponding water hump.
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Fluid-structure interaction
Figure 3.11. Solitary wave; the parameters are roughly in agreement with the observations of Scott Russel, as reported in [FAB 01]
3.2.7
Wave impacting a rigid wall
Figure 3.12. Wave impacting against a rigid wall, principle of the model
As already pointed out in subsection 3.2.4, when a wave approaches the coast, its height grows and at the encounter of a breakwater, a pier for instance, a
Surface waves
173
substantial part of the energy and momentum carried by the wave can be released with disastrous consequences for the structure integrity. Though this kind of impact problem is highly nonlinear in nature and difficult to describe accurately, a very simplified model can still be proposed to give a gross estimation of the transient loads exerted on the solid, in relation to a few properties of the incident wave, see [COO 95]. As in any impact problem, the shock force depends on three distinct quantities namely, the equivalent mass M e of material involved in the impact, the shock duration τ and the speeds U − and U + of the material just before and just after the shock respectively. Accordingly, the typical magnitude of the shock force is Fshock M e U − − U + / τ . The first simplifying idea to determine the three quantities just mentioned is that because the compressibility of a liquid like water is very low, the shock duration must be short in comparison with the characteristic time of the structural response. Therefore, as long as τ remains shorter than the period of the first few modes of vibration of the solid, the loading by the impacting wave can be modelled as a “pressure impulse” defined as the action of the pressure p ( t ) : P=
z
τ
0
p(t )dt
[3.125]
Provided P is used instead of p, there is no need to determine τ . It is worth mentioning that quite similar ideas are extremely useful to model impacts between solids as described in [AXI 04] and [AXI 05]. To estimate the equivalent mass of impacting water it is necessary to specify the geometry of the problem. As sketched in Figure 3.12, the wave is assumed to impact a rigid and fixed vertical wall, which actually means that the wall motion during the impact is supposed negligible, at least to calculate the impact load. As shown in Figure 3.12b, the wave is modelled as a fluid layer rushing towards the obstacle at a uniform horizontal speed U − = U 0 ; the underlying water is supposed to remain still. The relative thickness η of the two fluid layers is clearly related to the height of the wave crest and the water depth. To simplify further the analysis, the impact is supposed perfectly inelastic, that is U + = 0 . Starting from the momentum equation of an inviscid and incompressible fluid moving at velocity U ( x, t ) : ∂U [3.126] ρf + ρ f U .gradU + grad p = 0 ∂t we show below that the convective inertia term can still be neglected with respect to the local inertia term provided the impact is sufficiently short. Let L denote the length scale of the variations in pressure and horizontal velocity field. In terms of orders of magnitude, we get:
174
ρf
Fluid-structure interaction
U0 U2 p + ρf 0 + = 0 τ L L
[3.127]
Therefore the convective term can be neglected as long as the following equivalent inequalities hold: ρf
L U0 U2 >> ρ f 0 ⇒ cψ >> U 0 τ L τ
[3.128]
By substituting cψ = gH for L /τ , with the aid of [3.127] and [3.128], U 0 is estimated as follows: ⎛ U ⎞ gZ 0 g ρ f U 0 cψ ⎜ 1 + 0 ⎟ + ρ f gZ 0 = 0 ⇒ U 0 = Z0 ⎜ ⎟ c c H ψ ⎠ ψ ⎝
[3.129]
Supposing for instance an impact lasting less than ten milliseconds, L larger than ten meters and H of the order of ten meters, the momentum equation [3.126] can be linearized as long as the wave height Z 0 is of the order, or less, than ten meters. Then, integrating the linearized momentum equation over the shock duration, we arrive at: τ
⌠ ⎮ ⎮ ⎮ ⎮ ⎮ ⌡0
⎛ ∂ U ⎞ + grad p ⎟ dt = 0 ⇒ ρ f U + − U − + grad P = 0 ⎜ρf ∂t ⎝ ⎠
(
)
[3.130]
where P(x,z;t) is the action of pressure, or pressure impulse. Thus for the completely inelastic impact, we obtain: grad P = ρ f U 0 [3.131] Finally, as the fluid is incompressible, the pressure impulse P(x,z;t) is found to be governed by the following boundary value problem: ∂ 2P ∂ 2P + =0 ∂ x2 ∂ z2 ∂P ∂z
P ( L, z ) = 0 ; P ( x,0) = 0 ; ∂P ∂x
= 0 if − H ≤ z ≤ −η H z =0
;
=0
[3.132]
z =− H
∂P ∂x
= ρ f U0
if η H − ≤ z ≤ 0
z =0
which is of the same type as those discussed in Chapter 2. On the other hand, L denotes here an horizontal scale factor such that if x > L, the fluid is practically not affected by the impact. Thus it is important not to confuse L and L . Whereas
Surface waves
175
L is related to the wavelength, L is related to the heap of water involved in the impact motion. At this step, the value of L still remains unknown. Gravity can be safely discarded during the impact as τ is so short that the Froude number is very large. The boundary value problem [3.132] can be solved analytically using the general method presented in Chapter 2. By separating the variables, one obtains:
P( x, z ) = A( x ) B( z ) ⇒ A( x ) = A+ e
kx
+ A− e
− kx
B′′ A ′′ = k2 ; = −k 2 B A ; B( z ) = B+ e
ikz
+ B− e
[3.133] − ikz
The homogeneous boundary conditions at z = 0 and z = -H on one hand, at x = 0 and x = L on the other, lead to the admissible shape functions: ⎛ λ (L − x) ⎞ ⎛ λn z ⎞ An ( x ) = sinh ⎜ n ⎟ ; Bn ( x ) = Cn sin ⎜ ⎟ H ⎝ H ⎠ ⎝ ⎠ ( 2n − 1) π = n 1, 2,... where λn = 2
[3.134]
In agreement with the superposition principle, the pressure impulse is written as: ⎛ λ (L − x) ⎞ ⎛λ z⎞ P ( x, z ) = ∑ Cn sin ⎜ n ⎟ sinh ⎜ n ⎟ H ⎝ H ⎠ n ≥1 ⎝ ⎠
[3.135]
Differentiating [3.135] with respect to x, we get: ∂P 1 =− ∂x H
∑λ C n ≥1
n
n
⎛ λ (L − x) ⎞ ⎛λ z⎞ sin ⎜ n ⎟ cosh ⎜ n ⎟ H ⎝ H ⎠ ⎝ ⎠
[3.136]
In particular at the wall (x = 0): ∂P ∂x
=− x =0
1 H
∑λ C n ≥1
n
n
⎛λ z⎞ ⎛λ L⎞ sin ⎜ n ⎟ cosh ⎜ n ⎟ ⎝ H ⎠ ⎝ H ⎠
[3.137]
Once more, we encounter the difficulty that the z dependency of [3.137] differs from that required for direct identification of the coefficients. At the wall, the derivative of the pressure impulse is shaped as the following rectangular pulse: ∂P ∂x
x =0
= ρ f U 0 {U ( − z ) − U ( −η H − z )}
[3.138]
Here again U denotes the Heaviside step function. Fortunately, it is possible to expand the pulse as the suitable Fourier series:
176
Fluid-structure interaction
∞ ⎛λ z⎞ ρ f U 0 ⎣⎡ U ( − z ) − U ( −η H + − z )⎦⎤ = ∑ a j sin ⎜ j ⎟ j =1 ⎝ H ⎠ −η H
aj = −
2ρ f U0 H
⌠ ⎮ ⎮ ⎮ ⎮ ⌡0
2ρ U ⎛λ z⎞ sin ⎜ j ⎟ dz = − f 0 1 − cos (ηλ j ) H λj ⎝ ⎠
(
)
[3.139]
The Fourier series [3.139] is shown in Figure 3.13 as computed using the fifty first harmonics. The series is found to converge uniformly except of course at the free surface z = 0. The Gibbs oscillations have little effect, if any, on the final result [3.142] which yields the total fluid momentum and the equivalent mass, since such quantities are obtained by integration over the fluid depth. In this respect, the calculation could be performed using a much smaller number of harmonics without diminishing greatly the degree of accuracy. Identifying terms by terms the series [3.137] and [3.139], the coefficients Cn are found to be: Cn =
2ρ f U0 H 2 n
λ
(1 − cosηλn ) ; cosh ( λn L / H )
n = 1, 2,...
[3.140]
Figure 3.13. Profile of the incident momentum density ρ f U 0 expanded in Fourier series
Finally the pressure impulse is expanded as: P ( x, z ) = 2 ρ f U 0 H ∑ n ≥1
2 n
λ
⎛ λ (L − x) ⎞ (1 − cosηλn ) ⎛λ z⎞ sin ⎜ n ⎟ sinh ⎜ n ⎟ cosh ( λn L / H ) ⎝ H ⎠ H ⎝ ⎠
[3.141]
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177
Figure 3.14. Vertical profile of the pressure impulse for a few values of the length parameter L/H (water depth represented on the ordinate axis and impulse on the abscissa axis)
Figure 3.15. Vertical profile of the pressure impulse for a few values of the thickness parameter η
Figure 3.14 shows the vertical profile of the pressure impulse at the impacted wall for η = 0.5 , and a few values of the length parameter L/H. The remarkable result is that provided L/H takes on values larger than about two, the profile is practically independent of L. This clearly indicates that the length of the fluid layer affected by the impact is about Lc = 2 H . Of course, the exact value of Lc can depend on that of the depth parameter η, which characterizes the thickness of the fluid layer in motion just before the impact. Figure 3.15 shows the profile of the
178
Fluid-structure interaction
pressure impulse P(0,z) as plotted for distinct values of η covering the whole range of possible variation. Essentially the same results were obtained by selecting either L/H = 4, or L/H = 10. As could be anticipated, if η is increased, so is the magnitude of the pressure load, while the maximum of the pressure impulse takes place at a greater depth. By integrating such profiles on a strip of wall of unit width, the resulting impulsive load per unit wall width is obtained. Moreover, this load can be expressed in terms of an equivalent mass of water Me which would impact the wall at the uniform horizontal velocity U0 : ⌠
M eU 0 = ⎮⎮
0
⌡− H
P(0, z )dz
[3.142]
If the equivalent mass is used as an intermediate result to compute the dynamical response of the wall, it can be useful to define a generalized impulse accounting for the mode shapes of the structure. For instance, considering the case of a rocking mode of the type: ϕ( z ) =
F H + zI H H K
−H≤z≤0
[3.143]
The generalized impulse is defined as: ⌠
0
M aU 0 = ⎮⎮ ϕ ( z ) P(0, z )dz
[3.144]
⌡− H
After a few elementary manipulations we arrive at: M a = 2ρ f H 2 ∑ n ≥1
(1 − cosηλn ) λn4
(λ − ( −1) ) n +1
n
[3.145]
The result [3.145] is illustrated in Figure 3.16, which refers to a hypothetical green sea of front length W = H = 10 m . The generalized mass is plotted versus the depth ratio. It turns out that if η exceeds a few tenths of H, the generalized mass is a substantial fraction of the scaling factor ρ f η HLcW . Assuming U 0 10 m/s the kinetic energy of the impacting heap of water is of the order of 30 MJoule, for comparison it may be noted that to have the same kinetic energy a 40 ton truck must rush at about 140 kilometre per hour!
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179
Figure 3.16. Generalized equivalent mass of the wave for a rocking motion of the wall
3.3. Surface tension 3.3.1
Capillary waves, or ripples
In most applications of mechanical engineering, surface tension is far too small to be of practical importance. However, it is still interesting to describe briefly a few peculiarities of capillary waves. In general, they are coupled with gravity waves and called ripples. To demonstrate their major properties, it will suffice to consider the special case of one-dimensional propagation, just as we did for the gravity waves. The problem is formulated as follows: ∂ 2p ∂ 2p + =0 ∂ x2 ∂ z2 ⎡ ∂p σ f ∂ 3p ⎤ g p + − =0 ⎢ ⎥ ρ f ∂ x∂ z 2 ⎦⎥ z = H ⎣⎢ ∂ z
[3.146]
Once more, we seek solutions of the travelling type [3.4]. Furthermore, restricting the presentation to the deep water case, the vertical profile is found to be of the exponentially evanescent type: p(z) = p± e − k z
[3.147]
The dispersion equation is: ω 2 = gk +
σf ρf
k3
[3.148]
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Fluid-structure interaction
The phase and group velocities follow as: ⎛σ k g ⎞ cψ = ⎜ f + ⎟ ; ⎜ ρ k ⎟⎠ ⎝ f
cg =
ρ f g + 3σ f k 2 ρ f 2
(ρ
3 f gk + σ f k )
[3.149]
In Figure 3.17, the phase velocity is plotted versus the wavelength. The curve has a well marked minimum at the value: cm = 2 gσ f / ρ f ;
λm = 2π σ f / ρ f g
[3.150]
for water at 20°C, cm 23 cm/s and λm 1.7 cm . In the range of wavelengths larger than λm , the surface waves are dominated by gravity, while in the range of wavelengths smaller than λm , the surface waves are dominated by surface tension. In the extreme case where gravity can be neglected, the remarkable result is found that, in contrast with the case of gravity waves, the group velocity is larger than the phase velocity and both of them are a decreasing function of the wavelength: ⎛ 2πσ f cψ = ⎜ ⎜ ρ λ ⎝ f
⎞ ⎟⎟ ⎠
;
cg =
3 cψ 2
Figure 3.17. Phase velocity of the surface waves on deep water at 20°C
[3.151]
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181
Figure 3.18. Frequency of the capillary waves on deep water at 20°C
As a well known consequence, the wake around an obstacle moving with respect to the liquid, or fixed in streaming water, largely differs depending on the size of the solid. If it is smaller than λm , which is typically the case of a small insect, or a herbaceous plant stem, the wake is observed to develop a little upstream of the solid, while in the case of a larger obstacle, the wake is initiated directly at the solid wall. Furthermore, it can be also observed that the general shape of the wake also differs largely depending on whether the waves are dominated by gravity or by surface tension. For the analysis of such wakes, the reader is referred to [LAM 32]. Finally, the frequency of the ripples is plotted versus the wavelength in Figure 3.18 for water at 20°C. As expected, the frequency range is much larger than that of the gravity waves. 3.3.2
Surface tension and cavitation
3.3.2.1 Static equilibrium of a micro-bubble, or cavitation nucleus As already mentioned in Chapter 1 subsection 1.2.2.6, cavitation occurs in a liquid if the absolute pressure PL becomes less than a certain critical value denoted PC . Experiment shows that PC can be significantly less than the saturating vapour pressure PV . To explain the phenomenon, one is led to assume that a liquid contains always a certain number of very tiny gas bubbles, whose radius can be as small as a few micrometers and even less. Actually, the presence of such bubbles is also confirmed by experiment. They act as cavitation nuclei which are activated to initiate vaporisation of the liquid when PL is decreased, and which collapse in a very short time when PL increases again. The potential energy of a micro-bubble of
182
Fluid-structure interaction
initial radius R0 is obtained by adding to the gas and liquid terms [2.75], [2.76] already calculated in Chapter 2, the vapour term and the surface tension term, the later being of paramount importance at such small length scales. Gravity can be safely neglected because it is related to the ascending motion of the bubble due to buoyancy whose time scale is much larger than those considered here. Potential energy is written as a function of the reduced radial position r = R / R0 :
( P − P ) r 3 ⎛ σ f ⎞ 2 ⎪⎫ ⎪⎧ P 3 1−γ Ep ( r ) = 4π R03 ⎨ G 0 r ( ) + L V +⎜ ⎟r ⎬ 3 ⎝ R0 ⎠ ⎪⎭ ⎩⎪ 3 (γ − 1)
[3.152]
where it is assumed that γ differs from unity, though adaptation of the formula to the isothermal case would be straightforward, and the results similar. The static equilibrium radii re of the bubble at these pressures, if they exist, are the roots of the following equation: ∂Ep ∂r
r = re
⎛ 2σ ⎪⎧ 2 − 3γ = 4π R03 ⎨ − PG 0 re( ) + ( PL − PV ) re2 + ⎜ f ⎝ R0 ⎩⎪
⎞ ⎪⎫ ⎟ re ⎬ = 0 ⎠ ⎭⎪
[3.153]
Stability of the equilibrium at re is given by the sign of: ∂ 2Ep ∂r
2 r = re
⎧⎪ ⎛ 2σ ⎞ ⎫⎪ 1− 3γ = 4π R03 ⎨( 3γ − 2 ) PG 0 re( ) + 2 ( PL − PV ) re + ⎜ f ⎟ ⎬ = 0 ⎪⎩ ⎝ R0 ⎠ ⎪⎭
[3.154]
From [3.153], the gas pressure PG 0 in equilibrium with a reference external pressure PL 0 , for a bubble of reference radius R0 is: PG 0 = PL 0 − PV +
2σ f
[3.155]
R0
If the external pressure evolves, the equilibrium radius of the bubble changes, so that in the new equilibrium condition we get, after a few manipulations: PGe = PLe − PV +
2σ f Re
⎛R ⎞ with PGe = PG 0 ⎜ 0 ⎟ ⎝ Re ⎠
3γ
[3.156]
From [3.155] and [3.156], we obtain the equilibrium condition relating PLe and Re : PLe − PV +
2σ f
2σ f ⎞ ⎛ Re ⎞ ⎛ − ⎜ PL 0 − PV + ⎟⎜ ⎟ Re ⎝ R0 ⎠ ⎝ R0 ⎠
−3γ
=0
[3.157]
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183
For instance, let us consider a bubble immersed in water at θ = 20°C . Vapour pressure is PV = 0.023 bar . Surface tension at the bubble interface is σ f = 0.0726 N/m . If the radius of the bubble equilibrated by the liquid pressure PL 0 = 1bar is assumed to be R0 = 1 μm , then from [3.155] the gas pressure is found
to be PG 0 = 2.43bar . On the other hand, if the radius of the bubble at PL 0 = 1bar is R0 = 10 μm , the reference gas pressure will be lower, at PG 0 = 1.12 bar . Starting from an equilibrium reference state PL 0 ( R0 ) , we are interested first to study the evolution of the static equilibrium of the bubble when PL is reduced from the initial value. The plots shown in Figure 3.19 display the equilibrium conditions PLe ( Re ) , computed from [3.157] (assuming γ = 1.41 ), respectively for two bubbles with reference radius R0 = 1 μm and 10 μm . These curves may be conveniently analysed by using the potential energy [3.152]. This is illustrated in Figure 3.20, which shows the change of potential energy as a function of the bubble radius, at several values of the external pressure PL , for the bubbles addressed in Figure 3.19. In contrast with the potential energy plots of Figure 2.10, which have a single minimum except for the asymptotic case PL = 0 , those of Figure 3.20 show the existence of three distinct pressure ranges. First, if PL remains sufficiently large ( PL > PV ) there is a single radius of equilibrium, which is stable (plots with
PL = 3104 and 105 Pa , for both bubbles). In this range the value of Re steadily increases as PL decreases. Then, in an intermediate pressure range PLc < PL < PV , two equilibrium radii exist. Figure 3.20 indicates that the smaller radius is stable, but the largest one is unstable (plots with PL = −5104 , −2 104 and 0 Pa for the 1 μm bubble, and with PL = 0 Pa for the 10 μm bubble). Finally, if PL is further diminished below a certain threshold PLc , no static equilibrium exists anymore (plot with PL = −105 Pa for the 1 μm bubble, and with PL = −105 , −5104 and − 2 104 Pa for the 10 μm bubble). Physically, liquid vaporises in this range and the bubble expands, in principle without limit. From the curves shown in Figures 3.19 and 3.20, it is clear that smaller bubbles can stand lower pressures in the surrounding liquid without cavitating. Indeed, for the 1 μm bubble
PLc = −5.8104 Pa , while the critical pressure is only −1.5103 Pa for the 10 μm bubble. The corresponding critical radius are respectively Rc = 1.83 μ m and 29.4 μ m . The values of PL marking the limits of the intermediate range can be understood as follows. Equation [3.156] may be recalled to determine the liquid pressure which is needed to equilibrate a bubble of radius Re :
184
Fluid-structure interaction 3γ
2σ f ⎛R ⎞ PLe = PG 0 ⎜ 0 ⎟ + PV − Re ⎝ Re ⎠
[3.158]
Figure 3.19. Equilibrium pressure PLe as a function of the equilibrium bubble radius Re for two bubbles respectively with reference radius 1 μ m and 10 μ m at 1 bar
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185
It is noted that when Re tends to infinity, PLe tends to PV . Therefore, the saturating vapour pressure stands for the boundary between the high pressure and the intermediate pressure domains. At lower values of PL , as illustrated in Figure 3.20, the function has a single minimum which defines the radius Rc related to the smallest value PLc of liquid pressure compatible with a bubble in equilibrium. Rc verifies the equation: ∂PLe ∂Re
= −3γ PG 0 R03γ Re Re = Rc
− ( 3γ +1)
+
2σ f Re2
=0
[3.159]
Figure 3.20. Potential energy as a function of the bubble radius for several values of PLe for two bubbles respectively with reference radius 1 μ m and 10 μ m at 1 bar
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Fluid-structure interaction
Accounting for [3.158], the solution defining the critical point is: 1
⎛ 3γ PG 0 R0 ⎞ 3γ −1 2σ f ⎛ 3γ − 1 ⎞ ; PLc = PV − Rc = R0 ⎜ ⎟⎟ ⎜ ⎟ ⎜ 2σ Rc ⎝ 3γ ⎠ f ⎝ ⎠
[3.160]
Figure 3.21. Potential energy as a function of the bubble radius for the critical pressure PLc for the bubble with reference radius 1 μ m at 1 bar
It is important to notice that PLc is less than the vapour pressure and can become negative, provided the size of the cavitation nuclei is small enough, which implies in practice a careful gas removal from the liquid. Then, in the range PL < PLc cavitation occurs. As could be anticipated, the potential curve referring to the critical pressure for cavitation PLc has a horizontal inflexion point at the equilibrium radius Rc , indicating that the equilibrium at such point is neutral. This is highlighted in Figure 3.21, which displays the potential energy as a function of the bubble radius at the critical pressure PLc , for the 1 μm bubble. 3.3.2.2 The collapse of cavitation bubbles The time-scale of an imploding cavity is so short that the motion is essentially inertia-controlled. We can, as a first approach, ignore any thermal effect or mass transfer by diffusion. Empty cavities will collapse completely, while vapour-filled bubbles will fully collapse if all vapour has time to condense. Gas-filled bubbles
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187
will partially collapse, as the motion will be cushioned in the final stages by compression of the gas. Here the simple case of a vapour-filled collapsing bubble will be addressed, following the treatment due to Rayleigh, which neglects the potential energy due to the surface tension and the internal incondensable gas [RAY 17]. Let us consider the radius R(t ) of a bubble with initial value Ri subjected to an external pressure PL , which will be assumed constant during the implosion time. Accordingly, using expression [2.73] for the kinetic energy and [3.152] for the potential energy (with σ f set to zero), the mechanical energy of the system simplifies to: 2πρ L R 3 R 2 +
4π ( PL − PV ) R 3 = E0 3
with E0 =
4π ( PL − PV ) Ri3 3
[3.161]
where E0 is the energy imparted initially to the bubble. Therefore: 2 ( PL − PV ) ⎛ Ri3 ⎞ R 2 = ⎜ 3 − 1⎟ ⇒ 3ρ L ⎝R ⎠
1 dr =− dt Ri
2 ( PL − PV ) ⎛ 1 ⎞ ⎜ 3 − 1⎟ 3ρ L r ⎝ ⎠
[3.162]
where r =R / Ri . Then, as shown in [AXI 04], we may compute the time-domain nonlinear oscillations of this conservative system by integrating [3.162]: 1
t ( r ) = Ri
⌠ ξ3 3ρ L ⎮ dξ ⎮ 2 ( PL − PV ) ⎮⌡r 1 − ξ 3
[3.163]
The Rayleigh collapse time TC is obtained when r → 0 : 1
TC = Ri
⌠ 3ρ L ξ3 ⎮ dξ ⎮ 2 ( PL − PV ) ⎮⌡0 1 − ξ 3
[3.164]
It turns out that the integral [3.164] can be expressed in terms of the beta or gamma functions (see for instance [ABR 84]): 1
⌠ ⎮ ⎮ ⎮ ⌡0
ξ3 1 ⎛ 1 5 ⎞ 1 Γ (1/ 2) Γ (5 / 6) dξ = B ⎜ , ⎟ = 0.747 3 1−ξ 3 ⎝2 6⎠ 3 Γ (4 / 3)
[3.165]
and we obtain: TC 0.915 Ri
ρL PL − PV
[3.166]
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Fluid-structure interaction
Figure 3.22. Time history of the inertial bubble collapse
Figure 3.23. Collapse velocity as a function of the bubble radius
Let us consider a bubble in equilibrium at vapour pressure PV = 2300 Pa and liquid pressure PL = 105 Pa . First, cavitation is initiated by reducing PL below PV .
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189
Then, when the bubble has a radius Ri = 1 mm , it is subjected to a step pressure function of magnitude PL − PV = 105 Pa . Figure 3.22 shows the time history R(t ) of the inertial collapse, computed from [3.163]. Also, using [3.162], Figure 3.23 presents the phase trajectory with velocity restricted to the subsonic range R ( t ) ≤ 1000 m/s . Because gas pressure and surface tension are neglected, it is found that velocity tends to infinity when the bubble is entirely collapsed. Physical validity of such a model is questionable. In reality, the kinetic energy is converted into elastic energy due to gas compressibility and surface tension. As a consequence, it can be shown that the bubble is collapsed to a minimum radius, approximately given as: 1
Rmin = R0 ( γ − 1)
1 3( γ −1)
⎛ Ri ⎞ γ −1 ⎜ ⎟ ⎝ R0 ⎠
[3.167]
where R0 is the equilibrium radius at pressure PL . Note that most bubble collapse motions become so fast that the gas behaviour is much closer to adiabatic than isothermal, therefore we may assume that γ > 1 . Because typical velocities of the collapsing bubble interface are not negligible when compared with the sound speed in the enclosing liquid, compressibility effects in the surrounding liquid should not be neglected in a more refined analysis of the problem. The major point remains that the collapse of cavitation bubbles delivers short lived and intense pressure pulses, which causes pitting of the metallic surfaces thus creating corrosion sites and violent transients in pipe systems, as further considered in Chapter 6, subsection 6.4.2. 3.3.2.3 Oscillations and activation of the cavitation nuclei In the absence of external excitation, a cavitation nucleus behaves as a nonlinear conservative oscillator, which can be conveniently studied by using the invariance of mechanical energy. Again, using [2.73] for the kinetic energy and [3.152] for the potential energy, the implicit equation of the phase trajectories is written as:
( P − P ) r 3 + ⎛ σ f ⎞ r 2 ⎪⎫ = E ⎪⎧ PGe 3 1−γ r ( ) + Le V 2πρ L Re5 r 3r 2 + 4π Re3 ⎨ ⎜ ⎟ ⎬ 0 3 ⎪⎩ 3 (γ − 1) ⎝ Re ⎠ ⎪⎭
[3.168]
which is a natural extension of [3.161]. Here r =R / Re , PGe = PG ( Re ) , PLe = PL ( Re ) and E0 is the initial energy of the bubble. Such a phase portrait is shown in the lower plot of Figure 3.24, where the arrows indicate the travel direction. These plots refers to the smaller bubble in Figures 3.19 and 3.20, with the following numerical values: R0 = 1 μ m at PL 0 = 1bar , Re = 1.4 μ m at PLe = −0.429 bar , ρ L = 1000 kg/m 3 , γ = 1.41 , σ f = 0.0726 N/m , PV = 2300 Pa .
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Fluid-structure interaction
In contrast with the phase portrait of Figure 2.16, not all the trajectories are closed and the open trajectories correspond to unlimited expansion of the bubble, which means the system becomes nonlinearly unstable provided E0 is sufficiently large.
Figure 3.24. Phase portrait of a micro-bubble in water for several energy levels
From the practical viewpoint, unlimited expansion of the bubble means that the liquid cavitates. These results are in complete agreement with the equilibrium PLe ( Re ) and potential energy E p ( R ) plots also shown in Figure 3.24.
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191
Figure 3.25. Nonlinear oscillation of a micro-bubble in water for several values of r0
The time-domain nonlinear oscillations of the bubble may be computed from the system energy, for given initial conditions, as explained earlier. Assuming r (0) = r0 and r(0) = 0 , equation [3.168] becomes:
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Fluid-structure interaction
σ P −P ⎪⎧ PGe ⎪⎫ 3 1−γ 3 1−γ r ( ) − r0 ( ) + Le V ( r 3 − r03 ) + f ( r 2 − r02 ) ⎬ = 0 2πρ L Re5 r 3 r2 + 4π Re3 ⎨ Re 3 ⎪⎩ 3 (γ − 1) ⎪⎭
(
)
[3.169] Whence: r
t(r) =
⌠ ⎮ ⎮ ⎮ ⎮ ⎮ ⌡0
dξ 2 ρ L Re2
3(1−γ ) 0
⎡ PGe r ⎢ ⎣ 3 (γ − 1)
−ξ ξ3
3(1−γ )
P − P r3 − ξ 3 σ r2 − ξ 2 ⎤ + Le V 0 3 + f 0 3 ⎥ Re ξ ξ 3 ⎦
[3.170]
which may be numerically solved with considerable precision. Figure 3.25 shows the bubble responses computed from [3.170], for several values of the initial radius perturbation r0 , using the same physical parameters as in Figure 3.24. For increasing initial compression of the bubble, nonlinear effects become of paramount importance. The nonlinear oscillations observed in the stable domain are marked by increasing differences in the nonlinear stiffness between the contracting and expanding phase of the bubble motion. The motion evolves much faster when the bubble radius is near the minimum value and much slower as it approaches the maximum value, which is easily explained as the effective bubble stiffness increases significantly when R is much smaller than Re . The motion period of the nonlinear motion also increases with the perturbation magnitude and, as expected, unstable behaviour is displayed at higher energy levels. Indeed, notice that the stable oscillation at r0 = 0.60 and the unstable motion when r0 = 0.55 are fully consistent respectively with the closed and open trajectories shown in Figure 3.24. 3.3.2.4 Rayleigh-Plesset equation Bubble vibrations forced by an external fluctuating pressure PL ( t ) are of major importance to understand cavitation. Specialized literature is particularly abundant on the subject. The interested reader can be referred in particular to [PLE 64], [BLA 86], [LEI 94] and [FRA 95]. They can be studied by using the so-called Rayleigh-Plesset equation [PLE 60]: 2σ f ⎞ −3γ ⎛ 2σ f ⎞ 3 ⎞ ⎛ ⎛ ρ f Re2 ⎜ rr + r2 ⎟ − ⎜ PLe − PV + ⎟ r + ⎜ PL (t ) − PV + ⎟=0 2 ⎠ ⎝ Re ⎠ Re r ⎠ ⎝ ⎝
[3.171]
which is easily derived starting from the Lagrangian of the bubble. Beyond the inertial term, one can recognise the stiffness-type terms already addressed in equations [3.153] and [3.157]. Dissipative effects have been neglected, at this stage.
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193
If the fluctuating pressure PL (t ) is referred to the equilibrium pressure PLe , then
we may write P ( e ) ( t ) = PLe − PL (t ) and [3.171] becomes:
2σ f ⎞ −3γ ⎛ 2σ f ⎞ 3 ⎞ ⎛ ⎛ (e) ρ f Re2 ⎜ rr + r2 ⎟ − ⎜ PLe − PV + ⎟ r + ⎜ PLe − PV + ⎟ = P (t ) 2 ⎠ ⎝ Re ⎠ Re r ⎠ ⎝ ⎝
[3.172]
It is instructive to discuss first the linearized version of [3.172] without the forcing term. For small vibrations, the dimensionless radius is conveniently written as r = 1 + η . By expanding [3.172] to first-order in η, the following equation is obtained: ⎧ 2σ ⎫ ρ f Re2η + ⎨ f ( 3γ − 1) + 3γ ( PLe − PV ) ⎬η = 0 ⎩ Re ⎭
[3.173]
Transformation of [3.173] into the canonical form of a prestressed harmonic oscillator (cf. [AXI 04], Chapter 5) is immediate: M η + {K e + K 0 }η = 0
[3.174]
where the oscillator coefficients are as follows: - Mass: M = 4πρ f Re3 . - Elastic stiffness: K e = 8πσ f ( 3γ − 1) .
[3.175]
- Prestress stiffness: K 0 = 12πγ Re ( PLe − PV )
The mass coefficient is the added mass [2.69] of the breathing mode, as discussed in subsection 2.2.3.1. The elastic coefficient K e is related to the surface tension and to the polytropic coefficient γ of the gas. K e is always positive, which indicates that surface tension is a stabilising force. The prestress coefficient K 0 is proportional to the difference between the static pressure in the liquid and the saturating vapour pressure inside the bubble. According to whether the difference is positive, or negative, the prestress stabilizes, or alternatively, destabilizes the bubble. The remarkable point, which could be anticipated with the aid of the static equilibrium analysis performed in subsection 3.3.2.1, is that the bubble buckles (buckling, or divergence instability) if the radius becomes larger than the critical value [3.160]. As a matter of fact, computing from [3.173] the value of Rc which cancels the resulting stiffness coefficient K e + K 0 of the oscillator, we obtain precisely the corresponding value of the critical liquid pressure PLe , also specified in [3.160]. The natural frequency of the bubble breathing mode versus the mean radius is easily obtained as:
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Fluid-structure interaction 1/ 2
⎤⎫ ⎪⎧ 1 ⎡ 2σ f ω =⎨ ( 3γ − 1) + ( PLe − PV ) 3γ ⎥ ⎪⎬ 2 ⎢ ⎪⎩ ρ f Re ⎣ Re ⎦ ⎪⎭
[3.176]
which is plotted in Figure 3.26, by assuming the volume change of air is either isothermal (dashed line), or adiabatic (full line). The plots refers to the normal conditions of pressure and temperature.
Figure 3.26. Natural frequency of the breathing mode of an air bubble in water (dashed line isothermal, full line adiabatic)
To study the nonlinear forced oscillations, the Rayleigh-Plesset equation [3.172] needs to be solved. It is worth noticing that the numerical integration using an explicit algorithm is delicate, as this equation contains a nonlinear term depending on velocity. Convergence requires the use of time steps much shorter than the linear threshold value for linear stability (cf. [AXI 04], Chapter 5). Therefore, it can be advantageous to use an implicit scheme instead of an explicit one, with internal iterations to treat the nonlinear forces, as explained just below. The Newmark implicit algorithm was already introduced in [AXI 04], although in a linear context, and is based on the following approximations for the velocity and displacement: h ( rn + rn +1 ) 2 h2 = rn + hrn + ( rn + rn +1 ) 4
rn +1 = rn + rn +1
[3.177]
the equilibrium condition being written at the (n + 1) -th time-step: 2σ f ⎞ 2σ f ⎞ ⎛ ⎛ 3 2⎞ −3γ ⎛ (e) ρ f Re2 ⎜ rn +1 rn +1 + ( rn +1 ) ⎟ − ⎜ PLe − PV + ⎟ ( rn +1 ) + ⎜ PLe − PV + ⎟ = Pn +1 2 Re ⎠ Re rn +1 ⎠ ⎝ ⎠ ⎝ ⎝
[3.178]
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195
and, by substituting [3.177] into [3.178], a nonlinear implicit equation in rn +1 is obtained: e F ⎡⎣ rn +1 ; rn , rn , rn , Pn(+1) ⎤⎦ = 0
[3.179]
rn , rn , rn as well as the external forcing function Pn(+e1) ≡ P ( e) (tn +1 ) are known where quantities. The algebraic equation [3.179] may be solved for rn +1 by a suitable iterative method. A good starting value for iterative schemes is the value of the acceleration computed at the previous time-step, x (0) ≡ rn(0) +1 = rn . After a converged value of rn +1 has been obtained, the velocity rn +1 and displacement rn +1 are computed from [3.177].
A digression for brief discussion of root-finding methods is appropriate here. Function [3.179] will be denoted F ( x ) in the following, to alleviate notation, rn +1 . The simplest approach is the where x stands for the radial acceleration bisection algorithm, based on bracketing the sought root inside successively narrower intervals x ∈ [ a, b ] , where F ( a ) and F ( b ) have opposite signs. If the
function is continuous, then at least one root must lie in the interval. Then [ a, b ] is bisected and one determines in which half of the interval the root lies, and the process goes on until a given tolerance is reached. This method is simple and quite robust, but slow, and is often used in conjunction with other algorithms to improve their reliability. Notice the bisection method cannot be easily generalized to higherdimensional problems. The Newton algorithm, which is much more effective, is based on a first-order Taylor development of the function F ( x ) centred at the previous iteration step, x = x (i −1) : F ( x ) F ( x (i −1) ) + ( x − x (i −1) ) F ′ ( x (i −1) ) ; F ′ ( x (i −1) ) ≡
dF dx
[3.180] x = x( i −1)
Then, the next estimate x (i ) is computed from [3.180] with F ( x ) = 0 : x (i ) = x (i −1) −
F ( x (i −1) )
F ′ ( x (i −1) )
[3.181]
where it is required that F ′ ( x (i −1) ) ≠ 0 (hence multiple roots must be dealt differently). The Newton method converges quite fast, but imposes the knowledge of analytical function derivatives, or otherwise their numerical computation at each
196
Fluid-structure interaction
iteration. This inconvenience may be circumvented replacing the tangent F ′ ( x ( i −1) ) by a secant approximation, based on the most recent iterations: F ′ ( x (i −1) )
F ( x (i −1) ) − F ( x (i − 2) )
[3.182]
x (i −1) − x (i − 2)
therefore iterations proceed as: x (i ) = x (i −1) −
(x
( i −1)
− x (i − 2) ) F ( x (i −1) )
[3.183]
F ( x (i −1) ) − F ( x (i − 2) )
The secant approach usually converges slower than the Newton method, but is computationally cheaper per iteration. Generalization of the Newton method to higher-dimensional problems is immediate:
[ X ](i ) = [ X ](i −1) − ⎡⎣ J mn ([ X ](i −1) ) ⎤⎦
−1
F
([ X ] ) ( i −1)
[3.184]
where J mn designates the Jacobian matrix with coefficients ∂Fm ∂xn . However, as pointed, the knowledge of many local system derivatives is awkward and expensive. To avoid this inconvenient, Quasi-Newton methods approximate the inverse of the local Jacobian matrix without explicit knowledge of the functions derivatives. Convergence may be accelerated by using a quadratic interpolation on F ( x) or,
more generally, by incorporating second-order information F ′′ ( x (i −1) ) through the so-called Hessian matrix. For details on these algorithms, as well as a discussion of delicate and pathological aspects in root-finding, the reader is referred to [FOR 76] or [PRE 92]. In order to illustrate the difficulties of the numerical solution of the Rayleigh-Plesset equation, we will solve [3.172] under unforced condition P ( e ) ( t ) = 0 , subjected to a given initial perturbation about the equilibrium state 1 μ m at 1 bar – which corresponds to the uppermost point of the equilibrium curve
PLe ( Re ) shown in the first plot of Figure 3.19. Numerical solutions produced by the explicit scheme of central differences and by the Newmark implicit algorithm will be shown. As discussed earlier, all dynamical solutions pertaining to perturbations of this equilibrium state are stable. In this example a severe initial bubble expansion of r0 = 3 is imposed.
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197
Figure 3.27. Nonlinear oscillation of a perturbed 1 μ m bubble in water at 1 bar for r0 = 3
The first plot in Figure 3.27, obtained using the direct integration method [3.170], may be taken as the reference solution. It shows that the dynamical response of the bubble consists on a steady and strongly nonlinear solution. It is clear that time-step integration algorithms may run into difficulties due to the very short time-scale during the compressive phase of the bubble response. The following plots show the numerical responses obtained using both time-step integration algorithms, for three increasing – but close – values of the integration time-step. One can notice that the implicit Newmark algorithm consistently produces robust results, which preserve the conservative nature of the model. On the contrary, if h > 2.0 10−10 s , the explicit method cannot cope with the compressive
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Fluid-structure interaction
phase in adequate manner, and random errors arise which catastrophically deteriorate the numerical solution. Obviously, a small enough time-step can be found which will produce adequate results. But the point is that unacceptable results may be obtained, even if a numerical instability is not always obvious. In the present example, the periodic nature of the unforced bubble motion renders errors easily detectable; however such is not the case under forced conditions, especially near the system stability boundaries. Indeed, under oscillatory excitations, the RayleighPlesset equation may lead to either periodic or chaotic solutions, stressing the need for reliable integration schemes. In the following computations, the Newmark method will be used. We will now show a few examples to highlight several aspects of the dynamical behaviour of bubbles when subjected to progressive or periodic pressure changes. In all these computations, the initial conditions are those of a bubble in equilibrium, with r0 = 1 and r0 = 0 , as well as PL (0) = Pe (or, referring to [3.172], P( e) ( 0 ) = 0 ). Again, the subject bubble will be the one with reference radius R0 = 1 μm at PL 0 = 1bar , shown in Figure 3.24, all computations being referred to the equilibrium condition Re = 1.4 μ m at PLe = −0.429 bar . In Figure 3.28 we show two examples of sudden increase of the pressure PL (t ) in the surrounding liquid, followed later by a reversion to the initial equilibrium value PLe . In the first computation the maximum value of the pressure excursion is
105 Pa , while in the second computation it is 106 Pa . Both computations induce vibratory regimes during the pressure increase, about the new and smaller equilibrium radius. Understandably, the higher pressure excursion of the second computation produces a highly nonlinear response at higher frequency. Then, after recovering the initial pressure, system inertia induces significant low frequency vibrations about the original equilibrium radius. Neither of these computations led to unstable phenomena, which can however be easily obtained if an adequate pressure decrease is imposed, as shown in Figure 3.29. The two computation results presented in Figure 3.29 pertain to slightly different negative pressure excursions, respectively −8 104 Pa and −105 Pa , of much lower duration than in the previous example. One can notice that in the first case, even if PL (t ) becomes lower than the critical value PLc = −5.8104 Pa , the duration of the bubble expansion phase is not large enough to trigger the unbounded response. Hence, after pressure recovering, the bubble collapses to a steady vibration about the original equilibrium radius. However, such is not the case if the negative pressure excursion goes slightly beyond, at −105 Pa , in which case the bubble radius becomes greater than the critical value when pressure recovers. Then, stabilization of the system at pressure PLe is no more possible.
Surface waves
Figure 3.28. Nonlinear oscillation of a micro-bubble in water for two cases of sudden increase in pressure PL (t ) reverting later to the initial value Pe
199
200
Fluid-structure interaction
Figure 3.29. Nonlinear oscillation of a micro-bubble in water for two cases of sudden decrease in pressure PL (t ) reverting later to the initial value Pe
The next computations briefly explore the effects of periodic oscillatory pressure excitations. Referring to formulation [3.172] of the Rayleigh-Plesset equation, the fluctuating pressure about the equilibrium value PLe is given as: P(
e)
( t ) = PS sin Ω t
[3.185]
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201
Figures 3.30 to 3.32 present the forced responses of the bubble, for increasing values of the imposed frequency. The excitation circular frequency Ω is referred to the resonance frequency ω of the linearized bubble, obtained from [3.176], at about 4.7 MHz. For each excitation frequency, the bubble responses to three increasing levels of the fluctuating pressure PS were computed. The lowest level, at PS / PLe = 0.2 , may excite near-linear oscillations (except when Ω ω ). The second amplitude, PS / PLe = 1 , is more than enough to excite nonlinear regimes, but the fluctuating pressure is always above the critical value PLc . Finally, at the highest level PS / PLe = 2 , the fluctuating pressure becomes lower than PLc during part of each excitation cycle. At Ω = 0.2 ω , the excitation frequency used in Figure 3.30 is much lower than the resonance frequency of the linearized bubble. For PS / PLe = 0.2 , the near-linear system displays time-scales of both the excitation and the bubble resonance. When PS / PLe = 1 nonlinear effects are apparent, with alternating lower-frequency and higher-frequency responses, following the expansion and compression half-cycles of the excitation. At high amplitude PS / PLe = 2 , depression exceeds the critical value PLc during a significant fraction of time and cavitation arises. Notice how the bubble displays increasingly large radius excursions, alternating with violent collapsing, in a seemingly chaotic manner. As pointed in subsection 3.1.1.2, this is a typical scenario capable of inducing severe cavitation damage in nearby structural surfaces, such as the blades of ship propellers. However, useful applications of cavitation phenomena exist, such as surface cleaning using ultrasonic-induced cavitation, or the destruction of kidney stones with focused ultrasonic pulses. Additionally, we should mention the fascinating phenomenon of sonoluminiscence, a light emission process associated with a form of bubble collapsing [LEI 94]. Figure 3.31 shows the results obtained when the excitation frequency is Ω = ω . Notice that, even at low excitation level PS / PLe = 0.2 , the resonating bubble response displays a significant vibratory amplitude. Therefore, nonlinear terms are non-negligible and the effective resonating frequency is a bit different from the linearized value ω , hence the beating behaviour. Increasing the excitation level to PS / PLe = 1 magnifies both effects and, as a result, the motion asymmetry increases and the beating period decreases. When PS / PLe = 2 , the bubble displays again high amplitude excursions and violent collapsing, however due to the higher excitation frequency the motion is now stable.
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Figure 3.30. Oscillations of a micro-bubble in water for increasing values of the oscillatory pressure amplitude PS at excitation frequency Ω = 0.2 ω
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203
Figure 3.31. Oscillations of a micro-bubble in water for increasing values of the oscillatory pressure amplitude PS at excitation frequency Ω = ω
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Figure 3.32. Oscillations of a micro-bubble in water for increasing values of the oscillatory pressure amplitude PS at excitation frequency Ω = 3 ω
Finally, Figure 3.32 shows the bubble response when the excitation frequency is rather higher than the bubble resonance frequency, at Ω = 3 ω . The bubble amplitude increases with the excitation level, but is always comparatively low and the motion stable. The oscillations are strongly controlled by the bubble resonating frequency, implying that the corresponding spectra presents a significant 1/3 subharmonic of the excitation frequency.
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205
3.4. Sloshing modes If a finite volume of liquid is bounded by motionless and reflecting walls it can oscillate according to stationary free surface waves called sloshing modes. The main features of such modes are described below based on a few analytical examples. As in the case of solids, it is of interest to consider first discrete systems and then continuous systems. 3.4.1
Discrete systems
3.4.1.1 U tube
Figure 3.33. Sloshing of a liquid contained in a U tube
The simplest conceivable sloshing system is sketched in Figure 3.33. The liquid is contained in a U-tube of uniform cross-sectional area S f and limited by the free surfaces denoted (A) and (B), respectively. Adopting the one-dimensional column fluid model, it is assumed that during fluid oscillations, the free surfaces remain horizontal. The vertical displacements are denoted Z A and Z B respectively and the fluid mass is M f = ρ f S f L where L is the length of the fluid column. Solving the problem by using the Lagrange equation is immediate. The free surface potential energy is: Ep =
1 1 ρ f gS f Z A2 + ρ f gS f Z B2 = ρ f gS f Z 2 2 2
[3.186]
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Fluid-structure interaction
where, due to fluid incompressibility, Z B = − Z A = Z . The kinetic energy is: Eκ =
1 M f Z 2 2
[3.187]
The Lagrange equation follows as: M f Z + 2 g ρ f S f Z = 0
[3.188]
The U-tube may be rightly viewed as the fluid counterpart of the structural harmonic oscillator. From the mathematical viewpoint it is similar to a mass-spring system, from the physical viewpoint however a marked analogy with the pendulum can also be invoked as in both cases the system is prestressed by gravity, which leads to a prestress potential energy and not an elastic energy as would be the case for a massspring system. The Newtonian approach to the problem is a little more lengthy but presents no difficulties. Considering for instance harmonic oscillations, they are governed by the following equations: d2p =0 ; ds 2
⎛ 2 ∂ p⎞ ⎜ −ω p + g ⎟ =0 ∂ z ⎠ A, B ⎝
[3.189]
where s is the curvilinear abscissa along the tube axis, conventionally taken as positive from (A) to (B). Accordingly, it follows that: ∂p ∂s
=− A
∂p ∂z
; A
∂p ∂s
= B
∂p ∂z
[3.190] B
Solving equation [3.189] is immediate: ∂ p pB − p A = ∂s L p − pA 2 −ω p A − g B =0 L p − pA −ω 2 pB + g B =0 L
[3.191]
Using the change of variable δ p = p B − p A , the following harmonic oscillator equation is readily obtained in terms of pressure: 2g δ p − ω 2δ p = 0 L
leading to the natural circular frequency:
[3.192]
Surface waves
ω sl =
2g L
207
[3.193]
Of course, the problem can be expressed in terms of fluid displacement as well as in terms of pressure. Substituting the free surface conditions p A = ρ f gZ A and pB = ρ f gZ B into [3.192] the spectral version of equation [3.188] is immediately
recovered. 3.4.1.2 Interconnected tanks Many systems of practical importance are basically of the same nature as the U-tube, eventually involving several degrees of freedom. This is the case in particular, of storage tanks interconnected by piping systems. A simple example is shown in Figure 3.34. The liquid volumes contained in two tanks of cross-sectional area S1 and S2 are connected through a pipe of cross-sectional area S p and length L. The first sloshing mode of the system corresponds to a uniform vertical oscillation of the free surfaces. As the fluid is incompressible, we still have to deal with a single degree of freedom system. It is found convenient to describe it by using the fluid displacement X in the connecting pipe. The vertical displacements of the free levels follows as: Z1 = −
SP X S1
; Z2 =
SP X S2
Figure 3.34. Interconnected storage tanks
The potential and kinetic energies follow immediately as:
[3.194]
208
Ep =
Fluid-structure interaction
ρf g 2
{S Z 1
2 1
+ S2 Z 22 } =
ρ f gS P ( S1 + S2 ) 2
2 S1S2
X2
ρ S ⎧⎪ ⎛ H H ⎞ ⎫⎪ S1 H1Z + S P LX + S2 H 2 Z } = f P ⎨ L + S P ⎜ 1 + 2 ⎟ ⎬ X 2 Eκ = { 2 2 ⎩⎪ ⎝ S1 S2 ⎠ ⎭⎪ ρf
2 1
2
[3.195]
2 2
whence the natural frequency of the first sloshing mode: f sl =
1 2π
gS P ( S1 + S2 ) LS1S2 + S P ( H1 S2 + H 2 S1 )
[3.196]
NOTE.– Use of a Lagrange multiplier As an exercise, it is also interesting to treat the holonomic relation [3.194] by the Lagrange multiplier method. With this purpose in mind, the connecting tube is mentally cut at mid-length to define two subsystems. Each one is described by using the vertical displacement of the free surface. In the absence of connection, one obtains the two following Lagrange functions: ⎧ ⎪ L1 = ⎪ ⎪ ⎨ ⎪ ⎪L2 = ⎪⎩
2 ⎞ ρ f ⎛⎛ S L⎛ S ⎞ ⎞ ⎜ ⎜ S1 H1 + P ⎜ 1 ⎟ ⎟ Z12 − gS1 Z12 ⎟ ⎟ 2 ⎜⎜ 2 ⎝ SP ⎠ ⎟ ⎠ ⎝⎝ ⎠ 2 ⎞ ρf ⎛⎛ S L⎛ S ⎞ ⎞ ⎜ ⎜ S2 H 2 + P ⎜ 2 ⎟ ⎟ Z 22 − gS2 Z 22 ⎟ ⎟ 2 ⎜⎜ 2 ⎝ SP ⎠ ⎟ ⎠ ⎝⎝ ⎠
[3.197]
The connecting relation can be written as: ρ f ( S1Z1 − S2 Z 2 ) = 0
[3.198]
whence the constrained Lagrangian: L′ =
2 2 ρ f ⎪⎧⎛ S P L ⎛ S1 ⎞ ⎞ 2 ⎛ S P L ⎛ S2 ⎞ ⎞ 2 ⎪⎫ ⎜ ⎟ ⎜ ⎨ S1 H1 + ⎜ ⎟ Z1 + ⎜ S2 H 2 + ⎜ ⎟ ⎟ Z2 ⎬ 2 ⎪⎜ 2 ⎝ SP ⎠ ⎟ 2 ⎝ SP ⎠ ⎟ ⎪ ⎠ ⎝ ⎠ ⎭ ⎩⎝ ρf g − {S1Z12 + S2 Z 22 } − Λρ f ( S1Z1 − S2 Z 2 ) 2
The Lagrange equations follow as:
[3.199]
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209
⎛ L S1 ⎞ ⎜ H1 + ⎟ Z1 + gZ1 + Λ = 0 2 SP ⎠ ⎝ ⎛ L S2 ⎞ ⎜ H2 + ⎟ Z 2 + gZ 2 − Λ = 0 2 SP ⎠ ⎝ S1Z1 − S2 Z 2 = 0
[3.200]
Λ can be easily eliminated between the two differential equations to produce finally the harmonic oscillator equation:
⎛ S1 S2 ⎜ S2 H1 + S1 H 2 + L SP ⎝
⎞ ⎟ Z1 + g (S1 + S2 )Z1 = 0 ⎠
[3.201]
from which the natural frequency [3.196] is recovered. The Lagrange multiplier is found to be: ⎛ LS ( S − S 2 ) / 2 + S P S1 ( H1 − H 2 ) ⎞ Λ = ⎜⎜ 1 1 ⎟⎟ gZ1 S P ( S 2 H1 + S1 H 2 ) + LS1 S2 ⎝ ⎠
[3.202]
The constraint force related to the connection condition is ρ f S p Λ . 3.4.2
Continuous systems
The sloshing modes described in the last subsection correspond to the particular family of the non deformed free surfaces modes. As the liquid provided with a prestressed free surface is a continuous medium, there exist an infinity of other sloshing modes marked by a deformation of the free surface. Their essential features can be displayed in an example easily solved analytically. 3.4.2.1 Rectangular tank Let us consider the rectangular swimming pool shown in Figure 3.35. The modal problem is written as: ∂ 2p ∂ 2p ∂ 2p + + =0 ∂ x2 ∂ y2 ∂ z2 ∂p ∂x g
= x =0
∂p ∂z
∂p ∂x
= x=L
∂p ∂y
= y =0
− ω 2 p ( x, y,H ) = 0 z=H
∂p ∂y
= y =
∂p ∂z
=0 z =0
[3.203]
210
Fluid-structure interaction
L is the length of the pool, the width and H the depth of water. Once again, it is straightforward to solve [3.203] by separating the variables. The mode shapes are found to be: ϕ n,m ( x, y,z ) = cos
nπ x mπ y cosh zkn,m cos L cosh Hkn,m
[3.204]
where kn ,m is the modal wave number: 1/2
kn,m
⎛ n 2 m2 ⎞ = π ⎜ 2 + 2 ⎟ n = 0,1,2,... m = 0,1,2,... ⎠ ⎝L
[3.205]
Figure 3.35. Sloshing modes of a rectangular basin
Since the liquid volume must remain unchanged, n and m cannot be both equal to zero. The natural pulsations are: ωn,m =
( gk ) tanh Hk n,m
n,m
[3.206]
In a shallow pool, the natural frequencies increase as the square root of the water depth: f n,m =
gH 2
n2 m2 + L2 2
while in a deep pool it becomes depth independent:
[3.207]
Surface waves
fn,m =
F GH
1 g n2 m2 + 2 π L2 2
I JK
211
1/4
[3.208]
Such results show that, in a pool of rectangular cross-section, the modal density of the sloshing modes is higher than that of the flexure modes of a rectangular plate (cf. [AXI 05], Chapter 6) and higher in the deep water case than in the shallow water case.
Figure 3.36. Natural frequency of the first sloshing modes versus pool’s depth
Figure 3.37. Natural frequencies of the sloshing modes on deep water L = = 10 m
212
Fluid-structure interaction
n=1, m =0
n=1,m=1
n=3, m= 2 Figure 3.38. Mode shapes of a rectangular tank (see also colour plate 3)
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213
They are illustrated in Figures 3.36 to 3.38. In Figure 3.36, the natural frequencies of the sloshing modes n = 1, 2, 3, m = 0 are plotted versus the reduced depth H/L for a pool L = 25 m. Of course, the higher is n, the sooner is the deep water hypothesis verified, since the wavelength varies as 1/n. Correlatively, the thickness of the fluid layer affected by the sloshing motion also decreases as 1/n and the modal pressure is found to vary according the following vertical profile: − cosh π nz / L e cosh π nH / L
π n( H − z ) L
[3.209]
which brings out, for the same reasons, the characteristic length scale L / πn . In Figure 3.37, the natural frequencies of the sloshing modes of a deep pool L = = 10 m are plotted for m and n varying from 1 to 10. Modal density is nearly equal to hundred modes per Hz. Finally, Figure 3.38 displays a few mode shapes of the free surface. The wavy shapes are marked by antinodal lines at the solid walls and zero mean value when averaged over the unperturbed free surface. 3.4.2.2 Circular tank Using cylindrical coordinates, the modal problem is written as: ∂ 2 p 1 ∂p 1 ∂ 2 p ∂ 2 p + + + =0 ∂r 2 r ∂r r 2 ∂θ 2 ∂z 2 ∂p ∂p ⎛ ∂p ⎞ =0 ; =0 ; ⎜g −ω2 p ⎟ =0 ∂r r = R ∂z z = 0 ⎝ ∂z ⎠ z=H
Figure 3.39. Circular tank
[3.210]
214
Fluid-structure interaction
Once more, it can be solved by using the separation method. Thus p is assumed to be of the type: ⎧ sin nθ p ( r, θ , z ) = A ( r ) B ( z ) ⎨ ⎩cos nθ
[3.211]
Substituting [3.211] into equation [3.210] the following ordinary differential equations are obtained: d 2B − k 2B = 0 2 dz d 2 A 1 dA ⎛ 2 n 2 ⎞ + +⎜k − 2 ⎟ A = 0 dr 2 r dr ⎝ r ⎠
[3.212]
Using the axial boundary conditions, the following relations concerning the axial mode shapes and dispersion equation are obtained: B ( z , k ) = a cosh ( kz )
[3.213]
ω 2 = kg tanh ( kH )
On the other hand, by using the change of variable u = kr , A ( r ) is transformed into a ( u ) , which is governed by the standard Bessel equation (see Appendix A4): d 2 a 1 da ⎛ n 2 ⎞ + + ⎜1 − ⎟ a = 0 du 2 u du ⎝ u 2 ⎠
[3.214]
The general solution of [3.214] is in terms of Bessel functions of the first and second kinds. However, the last ones must be rejected since the elevation cannot tend to infinity on the cylinder axis. Therefore, the radial mode shapes are written as: An ( r ) = α n J n ( kr )
[3.215]
The modal wave numbers are given by the radial boundary condition, which reads as: ∂p = 0 ⇒ J n′ ( kR ) = 0 ∂r r = R
[3.216]
Substituting the m-th root α n ,m = Rkn ,m of equation [3.216], into the dispersion equation [3.213], the natural pulsations of the sloshing modes are found to be: ω n ,m = gkn ,m tanh Hkn ,m , where kn ,m =
α n ,m R
[3.217]
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215
The associated mode shapes can be written as: ⎧cos nθ ⎫ cosh k n ,m z ϕ n ,m ( r, θ , z ) = J n ( kn ,m r ) ⎨ ⎬ ⎩ sin nθ ⎭ cosh kn ,m H
[3.218]
Figure 3.40. Natural frequencies of the sloshing modes on deep water
R = 10 m ; H = 100 m
In Figure 3.40, the natural frequencies of the sloshing modes of a deep tank R = 10 m are plotted for m and n varying from 1 to 10. The modal density is essentially the same as in the case of the rectangular geometry. Figure 3.41 displays a few mode shapes of the free surface. Once more, the wavy shapes are marked by antinodal lines at the solid walls, a central node and zero mean value when averaged over the unperturbed free surface. On the other hand, orthogonality between mode shapes of distinct circumferential rank n is obvious. That between mode shapes of same rank n, and distinct radial rank m results from specific properties of the Bessel functions known as the Lommel integrals (see Appendix A4). Such properties can be used to expand functions defined over a finite interval as a series of Bessel functions. This kind of expansion is merely a particular case of modal expansions, as already discussed in detail in [AXI 05].
216
Fluid-structure interaction
Figure 3.41. Mode shapes: deformed free surface of a circular basin
3.5. Fluid-structure interaction 3.5.1
Coupling between sloshing and structural modes
When the liquid is limited by a free surface and flexible walls, the modal problem related to the system [3.1] reads as: ( K s − ω 2 M s ) X s = − pnδ ( r − r0 ) Δp = ρ f ω 2 X s .nδ ( r − r0 ) [3.219] ∂p ⎤ ⎡ 2 =0 ⎢⎣ −ω p(x, y,z) + g ∂z ⎥⎦ z=H
where the surface tension is discarded. The modal system [3.219] couples two continuous media, assumed to be enclosed within conservative boundaries. Thus, in the absence of coupling, the solid and the liquid oscillations can be described in
Surface waves
217
terms of structural modes of vibration and sloshing modes respectively. The fluidstructure interaction terms present in the left-hand side of [3.219], which are also conservative in nature, induce some coupling between such modes. The coupled natural modes of vibration are a compound of free oscillations of the fluid and the structure. It may easily anticipated that the practical importance of the coupling is controlled by the oscillatory Froude number F = ω 2 L / g . To be more specific, if ω refers to the vibration of the structure, in the range F much less than one, the gravity stiffness of the free surface largely prevails on the structural stiffness and the free surface acts as a fixed top. But if F is much higher than one, the gravity stiffness of the free surface is negligible in comparison with that of the vibrating structure, hence it can be neglected and the free surface acts as a pressure node. In both of these extreme cases, the fluid is coupled to the structural through inertia solely. However, inertia forces differ in both cases. Coupling between sloshing and structural modes is of practical importance in a limited range of F values near the Froude number Fs based on the sloshing pulsation. Though, the mechanism of such a coupling is not difficult to understand, the actual computation of such modes is generally a tedious task and a single analytical example will be described here, borrowed from [GIB 88]. As sketched in Figure 3.42, two circular cylindrical shells of revolution of radii R1 and R2 are set in a vertical coaxial configuration in such a way that the interspace is shaped as a coaxial annular space of thickness h, filled with water up to a height H. The bottom of the external shell is closed by a stiff plate, and the internal shell is also closed by a plate at each end. As a first simplifying assumption, the external shell is assumed to be fixed and the internal shell is supposed to vibrate according to a rigid mode of horizontal translation.
Figure 3.42. Concentric rigid shells delimiting a thin liquid layer with a free surface
The 3D modal problem reads as:
218
Fluid-structure interaction
⎧ ∂ 2 p 1∂ p 1 ∂ 2 p ∂ 2 p ⎪ + + + =0 ∂ r 2 r∂ r r 2 ∂ θ 2 ∂ z 2 ⎪ ⎪ ∂p ⎪∂ p =0 ; g = ω 2 p ( r,θ , H ) ⎨ ∂ z z=H ⎪ ∂ z z =0 ⎪ ∂p ⎪∂ p = ω 2 ρ f X 0 cos θ ; =0 ∂ r r=R ⎪⎩ ∂ r r = R 1 2
[3.220]
Solving [3.220] analytically is straightforward though rather tedious. Fortunately the calculation is largely simplified in the particular case of the thin layer model, without changing the physics of the problem. Hence, in what follows, it is assumed that: R2 R1 R
; R2 − R1 = h
; h/R << 1
[3.221]
Integration of the Laplacian reads as: R+h
⌠ ⎮ ⎮ ⎮ ⎮ ⌡R
⎧ ∂ 2 p 1∂ p p ∂ 2 p ⎫ − + ⎨ 2 + ⎬ dr = 0 r∂ r r 2 ∂ z 2 ⎭ ⎩∂ r
[3.222]
Once more the first term yields the source term of the averaged fluid equation: R+h
⌠ ⎮ ⎮ ⎮ ⎮ ⌡R
r=R+h
⎡∂ p ⎤ ∂ 2p dr = ⎢ ⎥ ∂ r2 ⎣ ∂ r ⎦r=R
= −ω 2 ρ f X 0 cos θ
[3.223]
The next two following terms can be treated together to give: R+h
⌠ ⎮ ⎮ ⎮ ⎮ ⌡R
r =R +h
⎛ p ⎞ ⎡ p⎤ d⎜ ⎟ =⎢ ⎥ ⎝ r ⎠ ⎣ r ⎦r=R
=−
h p (z) R2
[3.224]
where p ( z ) denotes the pressure averaged in the radial direction. Whence the onedimensional non-homogeneous fluid equation: h
d2 p h − p = ω 2ρ f X0 dz 2 R 2
[3.225]
The solution is found to be: p ( z ) = ae
−
z R
+ be
+
z R
−ω 2ρ f R
R X0 h
[3.226]
Surface waves
219
The integration constants are determined by using the vertical boundary conditions. After a few manipulations, the fluctuating pressure field can be finally written as: p ( z ) = ω2ρ f R
F cosh z/R R ⎛ ⎞ X0 ⎜ − 1⎟ h ⎝ F cosh H/R − sinh H/R ⎠
[3.227]
where F = ω 2 R / g is the Froude number appropriate to the present problem. The generalized force exerted by the fluid on the moving shell follows as: H
Q1 =
⌠ − ⎮⎮ ⌡0
p (z)
2π
⌠ X 0 dz ⎮⎮ ⌡0
( cosθ )
2
R dθ = − ω 2 ρ f π R2 H
R 2 X 0 ( β + 1) h
[3.228]
which is written in the more condensed form: Q1 = −ω 2 M a ( β + 1) X 02
[3.229]
In the specialized literature (see for instance [WEH 60], [HUL 82], [FAL 02]), a result of the type [3.229] is traditionally interpreted as defining an added mass coefficient of the kind M a′ (F ) = M a ( β + 1) which depends on the Froude number. As convenient as such a definition may be in engineering applications, it must be emphasized that the physics described by a mass coefficient differs profoundly depending whether it depends on frequency, or not. This because a mass coefficient depending on frequency can not describe an inertia force. This can be made particularly clear by shifting from the frequency to the time domain. Using the convolution theorem, the time history of the force [3.229] is found to be: ⌠
t
Q1 ( t ) = − M a X ( t ) − M a ⎮⎮ X (τ ) B ( t − τ ) dτ
[3.230]
⌡0
Where B stands here for the Fourier (or Laplace) transform of β (F ) . The convolution integral indicates the existence of retarded effects in the building up of the force exerted by the fluid on the solid. It can be anticipated that they are related to the wave propagation delays. However, for mathematical convenience, the properties of the coupling force are described by reverting to the frequency domain. The expression [3.229] brings out two distinct quantities of physical relevance, namely: 1. The added mass of the strip model: M a = ρ f π R2 H
R h
2. The dimensionless coupling parameter β (F sloshing modes:
[3.231]
)
between the structural and
220
β=
Fluid-structure interaction
RF H (1 − F / Fs )
[3.232]
It is noticed that β becomes infinite for the sloshing Froude number, which is found to be: Fs = tanh H/R
[3.233]
Indeed, the value of Fs corresponds to the natural pulsation of the first sloshing mode (n = 1) for which the free surface oscillates according to the vertical shape Z o cos θ . If the structure vibrates at the sloshing mode frequency ω s , an undamped resonant response is obtained. Such a feature clearly appears already in the coupling parameter β (F ) , plotted in Figure 3.43 for the particular case H/R = 3. As already mentioned just above, three distinct domains of response can be clearly identified, namely:
Figure 3.43. Coupling parameter β vs oscillatory Froude number H/R = 3, Fs 1
1 – Range of small Froude numbers: In the range F << Fs , β (F ) is nearly zero. As a consequence, the fluid force [3.229] reduces to the inertial force of the strip model. Such a result is natural since, at the frequency considered, the stiffness of the free surface is so large in comparison with that of the structure, that any vertical oscillation of the liquid is practically prevented.
Surface waves
221
2 – Resonant range: In the range F ≅ Fs , the stiffness provided by the fluid is of the same order of magnitude as that provided by the structure. In such a case, the interaction between the sloshing and the structural modes is important. To study the properties of the compound modes it is appropriate to solve the eigenvalues of the homogeneous equation:
(
)
⎡ K s − ω 2 M s + M a (1 + β (ω ) ) ⎤ X 0 = 0 ⎣ ⎦
[3.234]
3 – Range of large Froude numbers: In the range F >> Fs , The generalized force [3.229] tends to the asymptotic form: Q1 = −ω 2 M a X 02 where M a ρ f π R2
R⎛ R R ⎛H⎞ ⎛ H ⎞⎞ 1 − tanh ⎜ ⎟ ⎟ = ρ f π R 2 μ ⎜ ⎟ h ⎜⎝ H h ⎝R⎠ ⎝ R ⎠⎠
[3.235]
which means that as in the range of small Froude numbers, the fluid force is again practically inertial in nature. However, the added mass coefficient is lower than the value derived from the strip model, the weighting factor μ being less than unity, as shown in Figure 3.44. Such a result can be easily understood by considering that in the range F >> Fs the stiffness of the free surface is negligible in comparison with that of the structure. Therefore, the forces developed at the free surface to resist a vertical oscillation of the fluid are negligible. Hence, the fluid-structure coupling is again purely inertial in nature and the inertia force calculated by taking into account the free surface condition p(H) = 0 is less than that arising from the strip model, which tacitly assumes that the free surface is fixed in the vertical direction. Thus formula [3.235] provides the corrective factor μ ( H / R ) to be applied to the strip model in order to account for the pressure node at the free surface. It is worth noticing that the added mass coefficient differs by less than ten percent from the value issued from the strip model as soon as the aspect ratio becomes larger than ten. On the other hand, the present figure agrees satisfactorily with that obtained for a circular cylindrical shell of low aspect ratio, see Chapter 2 subsection 2.3.5.2.
222
Fluid-structure interaction
Figure 3.44. Free surface effect on the added mass coefficient
On the other hand, it is also worth stressing that the modal equation [3.234] differs from the canonical form ( K − ω 2 M ) ϕ = 0 . Therefore, it is not suitable for carrying out the standard eigenvalue and eigenvector analyses used in the framework of the discrete systems and structural elements (cf. [AXI 04] and [AXI 05]). A natural way to solve the type of modal problem which arises here, is certainly to follow the general modal testing method already outlined in [AXI 04] Chapter 9, which consists in analysing in terms of transfer functions, the forced responses of the coupled system, to a harmonic excitation of continuously varied frequency. Application of the method to the present problem is illustrated in Figures 3.45a,b,c by plotting the square modulus of the transfer function: H (ω ) =
⎡ K s + iω C − ω ⎣
2
(M
1 s
)
+ M a (1 + β (ω ) ) ⎤⎦
[3.236]
Depending whether gravity is neglected, or not, the plots are presented as a dashed or a full line. The numerical values used for the computation are as follows: R = 1 m ; H = 3m ; h = 10 cm ; e = 5 mm ; ρ s = 7 800 kg/m 3 ; ρ f = 1000 kg/m3 ς=
C = 0.01 (M s + M a )Ks
where e is the shell thickness and ζ stands for the modal damping ratio. Assuming a nonzero value for ζ is both realistic and convenient to avoid an infinite magnitude of response at resonance.
Surface waves
223
Figure 3.45a. Squared modulus of the transfer function ( F >> Fs )
Figure 3.45a illustrates the type of responses obtained in the high frequency range. By comparing the dashed and the full lines, it can be verified that the coupling between the sloshing mode and the structural mode is negligible, as they are practically superposed onto each other except of course near the sloshing mode peak.
Figure 3.45b. Squared modulus of the transfer function ( F Fs )
224
Fluid-structure interaction
Figure 3.45b illustrates the type of responses obtained in the resonant frequency range. Because of the coupling, the single resonance peak (dashed line) is split into two distinct resonance peaks situated on both sides of the dashed line peak.
Figure 3.45c. Squared modulus of the transfer function ( F << Fs )
Finally, Figure 3.45c refers to the low frequency range. Again, the coupling between the sloshing and the structural modes is rather poor, though still detectable. 3.5.2
Floating structures
3.5.2.1 Introduction The oscillatory motion of floating bodies is a subject of major concern in naval and off-shore engineering for various reasons related to the seakeeping performances of ships and floating platforms. The basic problem to be solved is to determine the pressure forces exerted on the body by a system of incident waves which travel in a practically infinite extent of liquid. At sea, the pressure field exerted on the body results generally from three distinct components related namely to the incident waves, the scattered or diffracted wave induced by the presence of the body considered as a fixed obstacle and, finally, the fluid-structure interaction component induced by the motion of the body. Furthermore, two types of wave body interaction problems are often encountered in practice, depending whether the cruise speed of the body is zero, or not. Actually, it turns out that theory is by far not simple from the mathematical standpoint, even if restricted to the simplest case of linear and harmonic oscillations at zero cruising speed. Its presentation is therefore beyond the purview of the present book and the reader interested to the subject is
Surface waves
225
reported to [WEH 71], [NEW 77] and to a few more recent papers such as [HUL 82], [DAM 00]. In a book of this nature it is nevertheless appropriate to present a short introduction to the subject based on a few drastically simplified models which are very helpful to highlight the major physical features of the problem, on a qualitative basis at least. The following subsections are concerned with the conservative aspects of the problem, damping related to radiation of water waves will be shortly addressed in Chapter 7. 3.5.2.2 Buoyancy of a boat
Figure 3.46. Cross-section of a floating boat, at static equilibrium on still water
Following here the synthetic presentation given in [SOM 50], we start from the cross-section normal to the longitudinal axis Oy (unit vector j ) of a floating boat which is represented in Figure 3.46. The water is still and the boat is in the state of static equilibrium. Ox (unit vector i ) denotes the horizontal axis defined as the intersection between the free surface and the plane of the cross-section. The vertical axis Oz (unit vector k ) is conventionally pointing toward the sea floor. The hull is modelled as a cylindrical shell with generating lines parallel to Oy. The centre of mass of the boat cross-section, or elementary strip of thickness dy, is denoted G while C denotes the centre of mass of the displaced water. Thus, G stands for the point of application of the weight, denoted W , while C stands for that of the buoyancy force, denoted B and defined as the resultant of all hydrostatic pressure forces acting upon the wetted wall of the hull. By carrying out the actual calculation, Archimedes’ principle is recovered and the reason why C is called the centre of buoyancy is made clear. The elementary pressure force is written as: dF = ρ f gzdsdyn [3.237] where ds is the line element of the cross-section and n the unit vector normal to the hull and pointing inward the boat. The elementary buoyancy force is the vertical
226
Fluid-structure interaction
component of dF . Vertical projection of the oriented area element yields the horizontal element area: dsdyn.k = dxdy [3.238]
Hence, the buoyancy force exerted on the hull is found to be: ⌠ B = − ρ f g ⎮⎮
⌡(S )
zdxdy = − ρ fV g
[3.239]
where V is the volume of the displaced water, or displacement. The formula [3.239] expresses Archimedes’ principle and implies that B is applied to the centre of gravity of the displaced water, just like the weight is applied to the centre of gravity of the solid body. The position of C is thus given by the barycentre coordinates: xc =
1 Aw
⌠ xdxdz ⎮ ⌡(SW )
;
zc =
1 Aw
⌠ zdxdz ⎮ ⌡(SW )
[3.240]
where (SW ) is the immersed part of the cross-section and Aw its area. In Figure 3.46 it is tacitly assumed that the cross-section and the ship’s loading are symmetrical with respect to the vertical plane containing Oz, hence xc = 0 . Static equilibrium implies that weight must be exactly balanced by buoyancy ( B + W = 0 ) and also that G and C must be located on the same vertical line, in such a way that the resulting torque is zero. This is precisely the case of the configuration depicted in Figure 3.46, where d stands for the depth of the wetted hull cross-section (ship draught) and b for the beam at waterline. 3.5.2.3 Stability of the static equilibrium An elementary and intuitive way to check the stability of the static equilibrium defined just above is to verify whether the perturbation forces and/or moments induced by any displacement, or rotation of the hull tend to bring the boat back to the static equilibrium state, or not. The rigid body motions of a floating ship have received specific names according to the degree of freedom involved, which are as follows: Xi : sway, Yj : surge, Zk : heave ψ x i : pitch, ψ y j : roll, ψ z k : yaw No hydrostatic pressure is induced by a sway, surge or yaw motions, hence static equilibrium is indifferent, or neutral, with respect to such motions. Heave implies a change in the volume of displaced water, hence in the buoyancy force, which
Surface waves
227
opposes the vertical displacement. Restricting the analysis to small displacements, the restoring force is easily found to be: Fz = − ρ f g A Zk = − K H Zk [3.241] where A is the area of the horizontal cross-section of the boat at the waterline. In the same way, pitch implies that the bow dips into the sea as the stern is lifted up from the sea. Accordingly, the buoyancy force is increased on the bow and decreased on the stern, the torque generated opposes the rotation in pitch. It is given by the following integral: ⌠ L M x = − ρ f gi ⎮⎮ y 2 b ( y )ψ x dy = − K Pψ x i
[3.242]
⌡0
where again b(y) is the beam of the boat at waterline. It can also be noticed that in reality coupling between pitch and heave modes is usually important since ship hulls are not symmetric fore and aft. Conditional stability occurs for the rolling motion, as shown schematically in Figure 3.47a,b. If the axis of symmetry of the boat cross-section is tilted by an angle ψ y = θ from the vertical direction, the shape of the volume of displaced water is changed and so is the position of the buoyancy centre, which is thus located on a curve C (θ ) . Restricting the analysis to small angular displacements, C (θ ) can be approximated by a circle arc, whose centre M c is called metacentre. Of course M c is located on the symmetry line. As made evident in Figures 3.47a,b, depending whether G is below, or above M c , the torque formed by W and B opposes the rotation in roll, or tends to increase it. The magnitude of the torque is: M y = HM s gθ j = − K Rθ j [3.243] where M s is the mass of the boat and H = M c G is the metacentric height, positive if G is above M c (unstable configuration) and negative if G is below M c (stable configuration).
228
Fluid-structure interaction
Figure 3.47. Stability of the static equilibrium:(a) stable and (b) unstable
3.5.2.4 Natural frequencies of the rigid body modes According to the linear seakeeping theory, the generalized pressure force exerted on the six degrees of freedom of a floating body is a linear functional of the displacement, velocity and acceleration components of the floating body motion, which can be written in the frequency domain as: 2 ⎣⎡Qw (ω ) ⎦⎤ = ⎡⎣[ K w ] − ω ⎣⎡ M w (ω ) ⎦⎤ + iω ⎣⎡Cw (ω ) ⎦⎤ ⎤⎦ [ a ]
[3.244]
where the displacement field is [ a ] = ⎡⎣ X Y Z ψ x ψ y ψ z ⎤⎦ . [ K w ] is the stiffness matrix induced by the buoyancy restoring forces which were addressed in the last subsection. As already mentioned in the context of the generalized force [3.229], in naval engineering ⎡⎣ M W (ω ) ⎤⎦ is termed “added-mass” matrix in T
Surface waves
229
spite of its frequency dependency. ⎡⎣CW (ω ) ⎤⎦ , which is also found to be frequency dependent, governs the energy dissipation due to the radiation of surface waves, as explained later in Chapter 7 and is therefore termed “damping matrix”. It turns out that theoretical determination of ⎡⎣ M W (ω )⎤⎦ and ⎡⎣CW (ω ) ⎤⎦ is by far not simple, as mentioned in the introduction to the subject. In most cases of practical interest they must be computed numerically. In agreement with formula [3.229], the results can be presented in a dimensionless form in which frequency is replaced by a characteristic Froude number F = ω 2 bm / 2 g , where bm stands for the maximum beam at waterline of the hull, or a closely related length. Provided the body oscillates at a fixed, or nearly fixed frequency ω0 , it remains possible to interpret ⎡⎣ M W (ω0 ) ⎤⎦ and ⎡⎣CW (ω0 )⎤⎦ as true added-mass matrix and viscous damping matrix, respectively. Moreover, as a general result, if F is sufficiently large, gravity effects become negligible and ⎡⎣ M W ( F )⎤⎦ tends to the true added-mass matrix which is computed using one of the methods described in Chapter 2, assuming a pressure node at the water-level. On the extreme opposite in the range F << 1 ⎡⎣ M W ( F )⎤⎦ tends to the true added-mass matrix as calculated by assuming a pressure antinode at the water-level. Nevertheless, in the elementary theory presented here, coupling between the floating body and the surface waves is neglected. Therefore, to determine the natural frequencies of the heave, pitch and rolling modes, the stiffness coefficients defined in the last subsection must be complemented with suitable data concerning the structural inertia and fluid added mass coefficients. According to the compilation presented in [BLE 79] they can be expressed as: Heave mode: Pitch mode: Rolling mode:
fH = fP =
1 2π
1 2π
fR =
1 2π
KH g 0.13 M sH + M aH d KP g 0.13 I sP + I aP d
[3.245]
KR gH 0.35 2 I sR + I aR bm
For convenience in the notation, the structural and fluid inertial coefficients are specified here by using a subscripted capital letter, the first appearing in the name of the oscillation mode. The exact value of the added mass coefficients depends on the shape of the hull and must be computed numerically in most cases of practical interest. However, as a gross approximation, fluid inertia coefficients of the same order of magnitude as the analogous structural coefficients can be adopted. According to such a very simplified model, the natural frequencies can be expressed
230
Fluid-structure interaction
as pendulum frequencies. Of course, the length of the equivalent pendulum is related to the geometry of the hull, described by the characteristic lengths d and bm already defined above. From such simplified formulas, it appears immediately that the frequencies of the rigid body modes of vibration lie in the spectral range of the wind induced waves, the swell in particular. Therefore, resonant response can be expected, which can be a nuisance for several reasons and even a safety issue due to the instability of the rolling mode, which can lead to capsizing, as further discussed in an analytical example in the next subsections. 3.5.2.5 Example 1: heave mode of a floating circular cylindrical buoy A circular cylindrical rod of length L and external radius R is assumed to float in dead water and is moored to the sea floor through a linear spring acting in the vertical direction, see Figure 3.48. Assuming that L is much larger than R, the fluidstructure interaction problem is treated by using a strip model, described here in the Oxz plane, where Ox is the horizontal axis and Oz the upward vertical axis. Accordingly all the force, stiffness and mass coefficients used to describe the system are “per unit cylinder length”.
Figure 3.48. Heave mode of a floating circular cylindrical rod (strip model)
Supposing that at static equilibrium the solid is half immersed, as sketched in Figure 3.49, for a vertical vibration Z 0 eiωt the coupled problem is governed by the following equations written in the frequency domain:
Surface waves
(K
− ω02 M s ) Z 0 = R ⎮⎮ ⌠
s
π /2
⌡−π / 2
Δp = 0 grad p.n
P
231
p cos θ dθ
[3.246]
= ω02 ρ f Z 0 cos θ
⎛ 2 ∂p⎞ =0 ⎜ −ω0 p + g ⎟ ∂ z ⎠ z=H ; x ≥R ⎝
where the first equation in [3.246] describes the heave mode of the floating cylinder, modelled as a conservative harmonic oscillator loaded by the fluctuating pressure at the wetted wall. In this equation, the stiffness coefficient K s per unit length is provided by the mooring device and is used as free parameter to let vary the natural frequency ω0 of the heave mode. The difficulty in solving [3.246] analytically stems from the boundary conditions which differ depending whether x is smaller or larger than R, precluding in particular the possibility for space variable separation. Actually, as shortly outlined later, in Chapter 5 in the context of sound waves and in Chapter 7 in that of water waves, the most general and efficient method to solve this kind of problems is based on a boundary integral formulation of the radiation and scattering problem. Moreover, in most cases of practical interest such integrals must be solved numerically by using the so called boundary element method, in short BEM. Once the pressure field has been determined by using the BEM (or another appropriate method), the heave mode is found to be governed by an equation of the type [3.244], written here as:
(K
s
(
+ K w + iωCW (F ) − ω02 M s + M a (1 + β ( F
)))) Z0 = 0
[3.247]
where the added mass and damping coefficients are function of the characteristic Froude number F = 2ω 2 R / g . K w is the buoyancy stiffness coefficient and M a the added mass coefficient at high frequency F >> 1 . As indicated in Figure 3.49, determination of K w is an elementary exercise. The hull is assumed to be pushed down by the quantity Z 0 under the waterline. The static pressure field related to the vertical displacement follows immediately as: p = ρ f gZ 0 cos θ
[3.248]
which leads to the restoring force: ⌠
Fz = ρ f gZ 0 ⎮⎮
π /2
⌡−π / 2
cos θ Rdθ = 2 ρ f gRZ 0
and finally to the buoyancy stiffness coefficient:
[3.249]
232
Fluid-structure interaction
Figure 3.49. Restoring force exerted on the hull
Kw =
Fz = 2 ρ f gR Z0
[3.250]
As the added mass coefficient is concerned to avoid the task of solving the system [3.246], we adopt the asymptotic value M a valid for F >> 1 , which means that a pressure node condition it is tacitly assumed at the water level. Starting from the simpler case of a cylinder in complete immersion in an infinite extent of fluid, the wall pressure associated with the inertial effect is approximated here as: ⎧ω 2 ρ f RZ 0 cosθ if -π / 2 ≤ θ ≤ π / 2 p ( R, θ ) = ⎨ 0 if π / 2 ≤ θ ≤ 3π / 2 ⎩
[3.251]
The added mass per unit cylinder length related to the field [3.251] is thus: Ma =
1 ρ f π R2 2
[3.252]
Such a result is broadly known as the “half-body approximation”. Moreover, in this particularly simple example, the added mass coefficient is equal to the physically displaced mass of fluid and M s = M a by virtue of Archimedes’s principle. The natural frequency of the floating cylinder is thus found to be: f0 =
1 2π
Ks + Kw ρ f π R2
[3.253]
Incidentally, by letting K s = 0 , it can be verified that the natural frequency becomes 0.127 g / R which agrees with the semi-empirical relationships [3.245].
Surface waves
233
Figure 3.50. Added mass coefficient computed by solving the radiation and scattering problem
Finally, the dimensionless added mass coefficient M w / M a = 1 + β ( F
)
is plotted in
Figure 3.50 by using the computed data published in [DAM 00]. The coefficient tends to infinity when F tends to zero, which means that the cylinder is blocked with respect to the waterline. This is a typical 2D effect which disappears in the case of 3D bodies. 3.5.2.6 Example 2: rectangular cross-section Let us consider a ship modelled as a homogeneous straight beam of length L, and constant rectangular cross-section, whose centre is the centre of gravity G, see Figure 3.51. The width (or beam) is denoted b, the height h and β = b / h is the aspect ratio of the cross-section. The equivalent structural density is ρ s = 500 kg/m 3 , so the depth of the immersed hull is d = h / 2 and the centre of buoyancy is at the centre of the immersed cross-section.
234
Fluid-structure interaction
Figure 3.51. Rectangular cross-section of the boat
Figure 3.52. Rolling mode
To study rolling, it is convenient to use two reference frames, Gxyz, fixed with respect to the waterline and Gξης fixed with respect to the boat, see Figure 3.52. Denoting θ the roll angle, the coordinates of the centre of buoyancy in the frame Gξης are: +b / 2
ξc =
2 hb
⌠ H /2 ⎮ ⌠ ⎮ ⎮ ⎮ ⎮ ⎮ ξ dξ ⎮ ⎮ ⎮ ⎮ ⎮ ⌡− ξ tan θ ⎮ ⎮ ⌡− b / 2
dς
=
bβ tan θ 6
[3.254]
+b / 2
2 ςc = hb
⌠ ⎮ ⎮ ⎮ ⎮ ⎮ ⎮ ⎮ ⎮ ⌡− b / 2
H /2
⌠ ⎮ ⎮ dξ ⎮ ς ⎮ ⎮ ⌡− ξ tan θ
dς
=
(
)
b ( β tan θ ) − 3 2
12 β
In the frame Gxyz, they are transformed into:
Surface waves
xc =
(
b sin θ β 2 ( 2 + tan 2 θ ) − 3 12 β
)
{(( β tanθ ) − 3) cosθ − 2β
b zc = 12 β
235
2
2
tan θ sin θ
}
[3.255]
The position of C is plotted in Figure 3.53 as a function of θ for a few values of the aspect ratio β. The roll angle is varied from zero up to θ max = tan −1 (1/ β ) , at which value the waterline reaches the deck of the boat. If xc (θ ) is negative, the configuration is clearly unstable, as the buoyancy force tends to increase the tilt angle. Therefore, it is found that there is a critical value β c below which the static equilibrium configuration θ e = 0 is unstable: β < βc =
3 1.225 2
[3.256]
The analysis in the nonlinear domain can be carried out by computing the potential of gravity as a function of θ. The momentum with respect to G is: M P (θ ) = M s gxc =
(
)
M s gB 2 sin θ ⎡ β 2 2 + ( tan θ ) − 3⎤ ⎣ ⎦ 12 β
[3.257]
Integrating [3.257] with respect to θ , the potential follows as: Ep (θ1 ) =
θ θ ⌠ 1 ⌠ 1 M s gB ⎪⎧ 2 ⎪⎫ 2 2 ⎨( 2 β − 3) ⎮⎮ sin θ dθ + β ⎮⎮ sin θ ( tanθ ) d α ⎬ ⇒ ⌡0 12 β ⎪⎩ ⌡0 ⎪⎭
⎛ (1 − cos θ1 )2 ⎞ ⎫⎪ M gB ⎧⎪ + 1⎟ ⎬ Ep (θ1 ) = s ⎨( 2 β 2 − 3) (1 − cos θ1 ) + β 2 ⎜ ⎜ cos θ1 ⎟ 12 β ⎪ ⎝ ⎠ ⎪⎭ ⎩
[3.258]
Leaving out the constant terms, the reduced potential is written as: ε P (θ ) =
Ep 12 M s gB β
= 3cos θ + β 2 tan θ sin θ
[3.259]
As shown in Figure 3.54, the potential [3.259] has a minimum if β is sufficiently large, as confirmed below. The angular position of static equilibrium verifies the following equation:
{
}
∂ εP 2 = ( 2 β 2 − 3) + ( β tan θ ) sin θ = 0 ∂θ
[3.260]
236
Fluid-structure interaction
Figure 3.53. Position of the centre of buoyancy in reduced coordinates
Figure 3.54. Reduced potential in the vicinity of the critical aspect ratio
The roots of [3.260], defined modulo kπ , are: θ1 = 0 ⎧ ⎪ ⎛ 3 − 2β 2 θe = ⎨ −1 = ± θ tan ⎜ 2 ⎪ ⎜ β2 ⎝ ⎩
⎞ ⎟ ⎟ ⎠
Stability depends on the sign of the second derivative at equilibrium:
[3.261]
Surface waves
∂ 2ε P ∂θ2
θ =θ e
{
}
= ( 2 β 2 − 3) + ( β tan θ e ) cos θ e + 2 β 2 2
( sin θe ) 3 ( cos θe )
237
2
[3.262]
It is thus concluded that the root θ e = 0 becomes unstable for β < β c = 3 / 2 . The non vanishing root, termed tilt angle, defines a second state of equilibrium, which is unconditionally stable. However, it is appropriate to check whether the tilt angle is less or greater than θ max . Such a constraint restricts the interest of the tilted equilibrium state to the fairly narrow range of aspect ratios 1 ≤ β ≤ 3 / 2 . In particular, if β = 1, the tilt angle is 45°. Restricting the analysis to the small angles about θ e = 0 , the kinetic energy per unit length of the boat strip is written as: Eκ =
2 −2 ⎛ bh 3 hb 3 ⎞ M s b ( β + 1) 1 2 Jθ ; J = ρs ⎜ + = ⎟ 2 12 ⎠ 12 ⎝ 12
[3.263]
The computation of the kinetic energy of the water set in motion is more difficult and this term is neglected here. The potential energy [3.259] is expanded to the second order in power of θ : Ep =
M s gb ⎛ 2 3 ⎞ 2 ⎜ β − ⎟θ 12 ⎝ 2⎠
[3.264]
whence the oscillator equation: M s b 2 ( β −2 + 1) 12
M gb θ + s ( 2 β 2 − 3) θ = 0 12 β
[3.265]
The natural frequency is: fR ( β ) =
1 2π
g bβ
⎛ 2β 2 − 3 ⎞ ⎜ −2 ⎟ ⎝ 1+ β ⎠
[3.266]
In Figure 3.55 f R is plotted versus β . Of course f R vanishes at β = β c . If β is less than β c , the boat capsizes. On the other hand, to shift the frequency range to values larger than 1 Hz, to avoid excitation by the swell and the discomfort of sea sickness, it would be necessary to select β values larger than about 40, which is unsuitable for an efficient navigation. As a consequence, it is preferred to damp out the rolling oscillations of the boat rather than to avoid resonant excitation. The point will be addressed shortly in Chapter 7.
238
Fluid-structure interaction
Figure 3.55. Natural frequency of rolling plotted versus the aspect ratio β = b / h
3.5.2.7 Rolling induced by the swell
Figure 3.56. Rolling excited by the swell
Let us model the swell as a sine wave, written as: 2π x ⎞ ⎛ Z ( x, t ) = Z 0 cos ⎜ Ω ( λ ) t − ⎟ λ ⎠ ⎝
[3.267]
On deep water, the frequency is related to the wavelength as follows: Ω (λ ) =
2π g λ
[3.268]
As a consequence, at frequencies below a few tens of Hz, λ is larger than a few ten meter (see Figure 3.5). The height of the waves being assumed to be of a few meter
Surface waves
239
only, the wave profile is tilted with respect to the horizontal direction by a small angle α related to the wave profile by: α=
dZ dx
= x =0
2π Z 0 sin Ω t λ
[3.269]
Using the linearized version of [3.257] it follows that the swell induces the torque: M P (θ ) =
M s gb ( 2 β 2 − 3) 12 β
(θ − α )
[3.270]
and the forced version of equation [3.265] reads as: ⎛ g ( 2 β 2 − 3) ⎞ ⎛ 2π Z ⎞ 0 ⎟ cos Ω t θ + ωR2θ = ⎜ ⎜ β ( β −2 + 1) b ⎟ ⎜⎝ λ ⎟⎠ ⎝ ⎠
[3.261]
As expected, resonant excitation of rolling occurs Ω = ω R . 3.5.2.8 Antiresonant absorber for rolling Figure 3.57 is a sketch of the device installed in the ship’s hull. It can be viewed as a U-tube closed at both ends. The area of the rectangular cross-section is S = a and length of the water column is LW = b + 2 H 0 . The remaining length h of the tube branches is filled with air at a pressure P0 , properly adjusted to tune the frequency of the sloshing mode to that of the rolling mode. As already explained in [AXI 04], Chapter 9, because of the coupling between the two modes, the undesired resonance is replaced by a pair of closely spaced resonances whose contribution to the response cancel each other on a fairly narrow frequency range centred at the natural frequency of the undesired resonance. Of course efficiency of such a device is essentially restricted to essentially monochromatic swell. As a final exercise in the present chapter, it is of interest to study the tuning of the sloshing mode by adjusting P0 and to establish the coupled equations of the sloshing and the rolling modes. The air column modifies the frequency of the sloshing mode by providing the following amount of elastic energy: Ee =
EG SZ 2 h
[3.272]
where EG is Young’s modulus of the gas and Z the magnitude of the displacement of the free surface. Using the relations [1.23] and the perfect gas law [1.30], the following remarkable result is easily obtained: EG = P0
[3.273]
240
Fluid-structure interaction
Figure 3.57. Sloshing resonator used as an antiresonant absorber
Pertinence of the isothermal, versus adiabatic expansion law shall be discussed in the next chapter. Calculation of the potential and kinetic energy of the sloshing mode presents no difficulty and is used to formulate the equation of motion as: P ⎞ ⎛ 2 ⎜ ρ L g + 0 ⎟ SZ + ρ L ( b + 2 H 0 ) SZ = 0 h ⎠ ⎝
[3.274]
where ρ L is the density of the liquid assumed to be much larger than that of the gas, allowing one to discard the gas inertia. The natural frequency of the sloshing mode immediately follows as:
f s ( P0 ) =
1 2π
P ⎞ ⎛ 2 ⎜ ρL g + 0 ⎟ h ⎠ ⎝ ρL (b + 2H0 )
[3.275]
The sloshing and the rolling modes are coupled together through the fluctuating pressure and motion at the bottom of the U-tube, below the free surfaces. Compressibility of water being discarded, the fluctuating pressure field is of the linear form p( s ; t ) = C1 s + C2 , where s designates the abscissa along the U-tube. As is rather obvious, the problem is skew-symmetric about the axis Oz. Hence taking the origin of s at the middle cross-section of the U-tube, the following boundary conditions must be verified: p(0; t ) = C2 = 0
;
P ⎞ b ⎞ ⎛ ⎛ p ⎜ H0 + ; t ⎟ = ⎜ ρL g + 0 ⎟ Z h⎠ 2 ⎝ ⎠ ⎝
[3.276]
Surface waves
241
The pressure field is thus: P ⎞ ⎛ 2s ⎞ ⎛ p( s ; t ) = ⎜ ρ L g + 0 ⎟ ⎜ ⎟ Z (t ) h ⎠ ⎝ b + 2H0 ⎠ ⎝
[3.277]
The pressure force exerted at the bottom of the branch s>0 is: ⌠
F1 = ⎮⎮
b/2
⌡b / 2 − a
p( s ; t ) ds
[3.278]
In the tube connecting the two branches, the force resultant is nil, since pressure is uniform in a same cross-section. Moreover, if a is much less than b, the integral [3.278] is practically equal to: b/2
P ⎞⎛ b ⎞ ⎛ p( s ; t ) ds = ap(b / 2; t ) = S ⎜ ρ L g + 0 ⎟ ⎜ ⎟ Z (t ) ⌡b / 2 − a h ⎠ ⎝ b + 2H 0 ⎠ ⎝ ⌠
⎮⎮
[3.279]
A fluctuating force is also exerted at the upper end of the tube branch, which is: P ⎞ ⎛ a p ( b / 2 + H 0 ; t ) = S ⎜ ρ L g + 0 ⎟ Z ( t ) h⎠ ⎝
[3.280]
Therefore, the resulting pressure force exerted on the right-hand side of the device is: P ⎞ ⎛ 2H0 ⎞ ⎛ a ( p(b / 2 + H ; t ) − p(b / 2) ) = − S ⎜ ρ L g + 0 ⎟ ⎜ ⎟ Z (t ) h ⎠ ⎝ b + 2H 0 ⎠ ⎝
[3.281]
Of course the resulting force on the hull is zero, since the pressure field is skewsymmetric. The resulting torque is: P ⎞ ⎛ 2 H 0b ⎞ ⎛ MR = −S ⎜ ρ L g + 0 ⎟ ⎜ ⎟ Z (t ) h ⎠ ⎝ b + 2H 0 ⎠ ⎝
[3.282]
As the problem is conservative, coupling must be symmetrical, hence the same torque must be exerted on the sloshing mode. This can be immediately verified by forming the functional of energy relative to the work of the torque for an angular displacement θ. The following form is obtained, which is symmetric with respect to Z and θ , as should be: P ⎞ ⎛ 2 H 0b ⎞ ⎛ TM = Mcθ = − S ⎜ ρ L g + 0 ⎟ ⎜ ⎟ Z (t )θ (t ) h ⎠ ⎝ b + 2H 0 ⎠ ⎝
The coupled equations of motion are:
[3.283]
242
Fluid-structure interaction
M S b2 ⎧ −2 g P0 ⎞ ⎛ 2bH 0 ⎞ ⎫ ⎛ 2 ⎨( β + 1) θ + ( 2 β − 3)θ ⎬ − S ⎜ ρ L g + ⎟ ⎜ ⎟ Z = F ( Ω ;t ) 12 ⎩ b h ⎠ ⎝ b + 2H 0 ⎠ ⎭ ⎝ P ⎞ P ⎞ ⎛ 2bH 0 ⎞ ⎛ ⎛ 2 ⎜ ρ L g + 0 ⎟ SZ + ρ L ( b + 2 H 0 ) SZ − S ⎜ ρ L g + 0 ⎟ ⎜ ⎟θ = 0 h ⎠ h ⎠ ⎝ b + 2H0 ⎠ ⎝ ⎝
[3.284]
Using [3.270], the load term induced by the monochromatic swell is: 2 M S b ⎛ g ( 2 β − 3) ⎞ ⎛ 2π Z 0 ⎞ ⎜ ⎟⎜ F (Ω ;t ) = cos ( Ω t ) ⎟ ⎝ λ ⎟⎠ 12 ⎜ β ⎝ ⎠
[3.285]
Chapter 4
Plane acoustical waves in pipe systems
Here, the fluid is supposed to be compressible and inviscid (non viscous). So, it is modelled as a continuous medium assumed to be elastic and conservative. Fluid elasticity combined with inertia creates the necessary conditions for the wavy nature of motions within the medium, called acoustic or sound waves. Hence, acoustics is often defined as that part of fluid dynamics which is concerned with compressible flows. However, to deal with fluid-structure coupled systems, it is appropriate to enlarge the definition as that part of continuum mechanics concerned with dilatation or pressure waves, independently of the solid or fluid nature of the continuum. Thus, the concepts and analytical tools already described in Volume 2 of this series in the context of elastodynamics in solids and structural elements - can be directly applied to acoustics. The analogy can be further enforced in the presentation by considering first the case of plane acoustic waves in pipes which stand for the fluid counterpart of the longitudinal vibrations in elastic beams. Despite of the analogy, fluid-borne sound is much more amenable to mathematical modelling than solid-borne sound, because in the last case dilatation waves are generally coupled to shear and bending waves, even if the problem is restricted to plane waves in beam structures. As described in the present chapter and later in chapters 6 and 7, plane sound waves in complex pipe systems can be analysed by using the so-called transfer matrix method (in short TMM) which is much more advantageous in terms of computational cost and accuracy than finite element method and even modal synthesis method, provided analysis is restricted to the linear domain. The chapter is concluded by describing the behaviour of the speed of sound in relation to the physical properties of the fluid and the elastic properties of the tube walls, including thermodynamic and nonhomogeneity effects.
244
Fluid-structure interaction
4.1. Introduction 4.1.1
Acoustics and sound perception
In acoustics, the spectral domain of practical interest is very extended. The human ear is sensitive to frequencies between about 25 to 30 000 Hz [SAU 92], [FLE 98]. Furthermore, the radiated power and the perceptible sound intensity (cf. formulas [3.28] and [3.29] in Chapter 3, which serve to define these quantities) can vary enormously from one case to another. Let us mention that the power level emitted by a bass drum can be as large as about 25 W; in contrast, the power emitted by a violin in the softest phrases is of the order of 410−6 W , [COT 90]. The threshold of perceptible sound intensity for a human ear is as low as 10−12 Wm -2 , whereas the threshold of pain, which leads to a fast and irreversible deterioration of the ear quality, is of the order of 1 Wm -2 . Furthermore, the progress achieved nowadays in the sensitivity of pressure sensors and in signal processing makes available to the engineer a much broader range of frequencies and intensities than that mentioned above. Acoustic engineering is concerned in particular with identification of various sources of noise (acoustic signature) and reduction of noise sources or transmission. On the other hand, ultrasonic waves have many applications in scanning and cleansing techniques for instance. Amongst many distinct fields of interest, musical acoustics fully deserves a particular mention, even if the subject remains rather peripheral from the engineering standpoint. Indeed, the practice of music and manufacturing of musical instruments is nearly as old as humanity. The most famous physicists and mathematicians devoted numerous important experimental and theoretical works to the subject. In fact, as emphasized in particular in [BEN 92] “a physicist is able to retrace a substantial part of mathematical physics while listening to a concert in relation both to the music itself and to the design of the musical instruments.” Although musical acoustics remains out of the scope of the present book, we shall touch on the topic to illustrate a few important aspects of linear acoustics in plane wave approximation. One reason not to linger further “to the delicious pleasure, each time renewed, of an unnecessary occupation” [RAV 11] is that even if engineering science is able to understand most of the physical mechanisms put to good use in the design of musical instruments, in the present state of the art it remains often less than the empirical knowledge accumulated by the music instrument makers. Actually, in musical acoustics nonlinear mechanisms are often of great importance; moreover, the ear is a very sensitive organ also able to interpret the received signal in such a way that “there exists a psychoacoustic relationship between the physical nature of the sound and the perceived result” [RIS 92]. As a consequence, many small manufacturing details used to build a musical instrument can make a large difference in the final product. The reader interested in this fascinating and rapidly evolving field of acoustics is referred in particular to [FLE 98].
Plane acoustical waves in pipe systems
4.1.2
245
Acoustics in the context of fluid-structure interaction
In a book of this nature, the presentation is necessarily restricted to those basic elements of acoustics the knowledge of which is needed to study fluid-structure coupled systems, the interest being more specifically focused on the response of the solid. In many applications, the first natural frequencies of the structures lie within the infrasonic or low sonic frequency range and, concomitantly, to that of the large wavelengths. Such a domain is generally well suited for a modal description of the solid and the fluid. This is also of practical interest as many external acoustical sources exist which can excite efficiently the first vibration modes of flexible structures. It is mainly to this kind of application that the fundamentals and basic formulations presented in this book are addressed. It is however necessary to stress that needs in vibroacoustics encountered in engineering often cover a higher spectral range as well, extending roughly from about a few hundred Hertz up to a few ten kilohertz. Such a range is generally marked by an extremely large density of vibroacoustic modes, making thus the use of modal synthesis methods much less attractive and even feasible than in the low frequency range. Practical interest in the high frequency range has increased significantly during the last two decades with the aim of reducing the noise level of various kinds of machineries and vehicles and improving the acoustic quality of concert halls. The reader interested in the topic may be referred in particular to [LES 88], [LYO 75], [FAH 01]. 4.1.3
Linear and conservative acoustical wave equation
As shown in Chapter 1, subsection 1.3.1, starting from the Euler equations linearized about the static state of equilibrium, the acoustical wave equation [1.57] is obtained, which is written here in Cartesian coordinates as: ∂2 p ∂2 p ∂2 p 1 ∂2 p e + + 2− 2 = S ( ) ( x , y , z; t ) 2 2 2 ∂x ∂y ∂z cf ∂ t
[4.1]
where S ( ) stands for an acoustical source term which will be specified later, in subsection 4.3 for the one-dimensional case and in Chapter 5 section 5.3, for the three-dimensional case. It is recalled that c f is the speed of sound defined in e
Chapter 1 as a mechanical property of the medium as: cf =
∂P = ∂ρ 0
1 ρ fκ f
[4.2]
where κ f is the coefficient of compressibility of the fluid. Similarity with the case of an elastic solid can be further highlighted by expressing the speed of sound as:
246
Fluid-structure interaction
cf =
Ef
⇔ Ef =
ρf
1 = ρ f c 2f κf
[4.3]
E f denotes Young’s modulus of the fluid.
Furthermore, equation [4.1] is basically the same as that which governs the elastic dilatation waves in a three-dimensional solid. As already mentioned, a direct consequence of such a close similarity is that the analytical tools used to describe elastic waves in solids are also available to study acoustics. Therefore, in this chapter and the next, the methods already presented in [AXI 05] will be briefly restated and then extended taking into account the specific concerns of practical importance in acoustics. As a preliminary, it is recalled that in the absence of external excitation, the harmonic solution of equation [4.1] can be obtained by the method of variables separation. Adapting the presentation made in [AXI 05] Chapter 1 to the present case, if Cartesian coordinates are used, the sound pressure field is written as the complex amplitude: p ( x, y , z; ω ) = eiωt p x ( x ) p y ( y ) pz ( z )
[4.4]
Substituting the assumed solution [4.4] into [4.1] it can be easily shown that the general solution is: p± e
i ωt ∓ k . r
(
)
= p± e
⎛ . r ⎞ iω ⎜ t ∓ ⎟ ⎜ cf ⎟ ⎝ ⎠
[4.5] where the Cartesian coordinates of r are x, y, z and those of the wave number vector k are denoted k x , k y and k z . The direction of wave propagation is specified by the unit vector = k / k . The wavelength is: 2π 2π λ= = cf k
[4.6]
The solution [4.5] stands for a pair of harmonic and travelling sound waves, p+ is termed a forward, or outgoing wave while p− is termed a backward, or incoming wave. Furthermore, if is constant the waves are plane, which implies that the phase angle of the complex amplitude is the same at any point of a same plane perpendicular to (or to k ) and so are pressure and velocity of fluid particles. As in the case of structural elements, it is found convenient to start the detailed presentation of sound waves by treating first the case of plane waves which, as a definition, depend on a single space coordinate, x if the direction of the Ox axis
Plane acoustical waves in pipe systems
247
coincides with . Furthermore, in the context of fluid-structure interaction problems, it is found especially interesting to particularize further the study to the plane waves in pipe systems. Such internal waves may be rightly considered as the fluid counterpart of the dilatation waves in solid beams and they are of practical concern in many fields of application, including musical and engineering acoustics.
In section 4.2, the travelling and standing sound waves are studied by using first the same analytical methods as those already described to study the natural modes of vibration of structural elements and then by introducing a distinct method known as the transfer matrix method. Section 4.3 is devoted to the waves excited by external acoustic sources. Again, the forced waves can be analysed by using various methods, including those commonly used in structural mechanics and the transfer matrix method. In section 4.4, a few physical models concerning the speed of sound in various materials are discussed. As a preliminary, the mechanical aspect of fluid compressibility is related qualitatively to interatomic (or intermolecular) forces, which largely differ depending whether the material is gaseous, liquid or solid. Then, the validity of the adiabatic versus the isothermal model for the speed of sound in gases is verified, based on the thermoacoustic characteristic times. Finally, modelling of sound speed is described, first in gas-liquid mixtures and then in liquids enclosed in ducts with elastic walls. 4.2. Free sound waves in pipe systems: plane and harmonic waves 4.2.1
Acoustic impedances and standing sound waves
4.2.1.1 Plane wave approximation in pipes The plane wave approximation extends to the case of a compressible fluid the column model already introduced in Chapter 2. It is recalled that, by definition, the volume offered to the fluid in a tube is characterized by one dimension L, termed the length, which is much larger than the other two, the so-called transverse dimensions, see Figure 4.1. As a consequence, for a fluid oscillation at a given frequency ω the compressibility parameter is much larger in the longitudinal direction, specified by the longitudinal unit vector ( s ) , than in any transverse direction. Moreover, if ω is sufficiently small, compressibility can be safely accounted for in the ( s ) direction solely. Hence, in a tube, the idea is to assume that pressure is constant over the tube cross-sections, so long as the wavelength is much larger than the transverse dimensions of the tube. Accordingly, the plane wave assumption is expected to be valid in the long wavelength range, and correlatively in the low frequency range such that:
248
Fluid-structure interaction
c λ >> 1 ⇔ f R ωR
[4.7]
Figure 4.1. Fluid column model
It is noticed that the dimensionless parameter appearing in the condition [4.7] is precisely the reciprocal of the compressibility parameter defined in Chapter 1, formula [1.107]. The spectral domain where fluid compressibility is significant in along the tube axis and negligible with regard to inertia in any cross-sectional direction follows immediately as: cf L
≤ ω <<
cf R
[4.8]
L is the tube length and R denotes the radius, or more generally, the largest crosssectional dimension of the tube. Validity of the plane wave approximation in pipes will be further addressed in Chapter 5 in the context of the three-dimensional guided waves. 4.2.1.2 Plane wave equations in pipes Taking into account fluid compressibility through the linear elastic law [1.32] and Young’s modulus [4.3], the one-dimensional fluid equations [2.10] become: iω dq Sf p + =0 ρ f c 2f ds iω q +
Sf d p =0 ρf ds
[4.9]
[4.10]
Plane acoustical waves in pipe systems
249
where q ( s ) eiω t , p ( s ) eiω t are the volume velocity and the pressure of the plane wave. Elimination of q between equations [4.9] and [4.10] is immediate. The result is the one-dimensional wave equation expressed in terms of pressure as: Sf d ⎛ S f dp ⎞ d ⎛ S dp ⎞ 2 S f 2 p=0⇔ ⎜ f p=0 ⎜⎜ ⎟⎟ + ω ⎟+k 2 ds ⎝ ρ f ds ⎠ ρ f cf ds ⎜⎝ ρ f ds ⎟⎠ ρf
[4.11]
where k = ω / c f is the wave number. Equation [4.11] is of the same type as that which describes the longitudinal waves in a straight beam, longitudinal displacement being replaced by pressure. On the other hand, according to equation [4.11], the plane sound waves are found to be independent of the tube curvature. Hence the waves in a tube of constant crosssectional area are the same, whether the tube is straight or curved. Amongst other applications, such a property is used in various musical wind instruments, the slidetrombone for instance, to devise tubes of suitable acoustical length and still reasonably sized for playing and transport. Hereafter, for convenience, we shall consider straight tubes only. 4.2.1.3 Travelling waves in a uniform tube and tube impedance
Figure 4.2. Example of a uniform tube
The simplest case, hereafter termed uniform tube for the sake of brevity, is that of a tube of constant cross-sectional area S f , filled with a homogeneous fluid ( ρ f and c f independent of position). In Figure 4.2, it is represented as a circular cylindrical tube of axis Ox, for convenience. The acoustical equation [4.11] simplifies into: d2p + k2 p = 0 dx 2
[4.12]
In agreement with the 3D solution [4.5], the 1D solution is written in terms of travelling waves as: p( x; k )eiωt = p+ e (
i ω t − kx )
+ p− e (
i ω t + kx )
[4.13]
250
Fluid-structure interaction
where p+ is the magnitude of a forward wave and p− that of a backward wave. With the aid of the momentum equation [4.10], the related volume velocity is: q( x; k )eiωt = ( q+ e − ikx − q− e + ikx ) eiωt =
Sf ρ f cf
(p e +
− ikx
− p− e + ikx ) eiωt
[4.14]
In these travelling waves, the pressure and volume velocity are related to each other by the following relations of proportionality: p+ =
ρ f cf Sf
q+
p− = −
;
ρ f cf Sf
q−
[4.15]
Referring to the concept of fluid impedance in a pipe, introduced in Chapter 2, subsection 2.2.2.4, the relations [4.15] can serve to define the forward and backward acoustic impedances which characterize the fluid contained within a pipe as the propagation of acoustical plane waves is concerned. They are expressed as: Z+ =
ρ f cf Sf
=Z
; Z− = −
ρ f cf Sf
= −Z
[4.16]
Going a little step further, if the volume velocity is replaced by the particle velocity in relations [4.16] the so-called specific impedance of the fluid is obtained: Z+(
sp )
= ρ f cf = Z (
sp )
; Z−(
sp )
= − ρ f c f = −Z (
sp )
[4.17]
In SI units, specific impedances are measured in Rayleighs. Depending on the nature of the elastic medium, the specific impedance can vary in enormous sp proportions, for instance in air at STP (20°C and 1 bar) Z ( ) 400 rayls whereas in steel Z (
sp )
4107 rayls , for other illustrative values see Appendix A6.
4.2.1.4 Reflected and transmitted waves at a change of impedance Let us consider a pipe made of two straight tube elements of distinct crosssectional area and filled with the same fluid, as shown in Figure 4.3. The tube impedance of the first element is Z1 = ρ f c f / S1 and that of the second element is Z2 = ρ f c f / S2 . As Z2 differs from Z1 , a forward pressure wave p+ experiences a
change of impedance when passing from tube (1) to tube (2). As a consequence, part of it, denoted ptr is transmitted while the other part, denoted p− , is reflected. To assess the efficiency of wave reflection and transmission one is led to define the refection coefficient R and the transmission coefficient T as: R=
p− p+
; T=
ptr p+
[4.18]
Plane acoustical waves in pipe systems
251
These coefficients are related to the impedances of the tubes as follows. In agreement with the continuity equations [2.28], it is easily checked that at the junction of the two tubes the following conditions must hold: q+ + q− = qtr
[4.19]
p+ + p− = ptr
Figure 4.3. Tube with a sudden cross sectional area change
In the jump conditions [4.19], the pressure equation states that the pressure is the same at both sides of the geometrical discontinuity, which is required for the equilibrium of the mass-less interface. The volume flow rate equation states that the flow at the outlet of the first tube is equal to the flow at the inlet of the second tube, which is necessary because in the absence of source or sink of fluid material at the interface, particle velocity must be continuous. The pressure equation implies that: 1+ R = T
[4.20]
With the aid of the impedances [4.16], the mass flow rate equation leads to: p+ p− ptr − = ⇔ (1 − R ) Z2 = T Z1 Z1 Z1 Z2
[4.21]
Equations [4.20] and [4.21] can be solved for R and T: R=
Z2 − Z1 Z1 + Z2
; T=
2Z2 Z1 + Z2
[4.22]
In fact, the results [4.22] are valid for any kind of impedance discontinuities, due to a change either in cross-sectional area, or fluid density and speed of sound. Conversely, if the change in these quantities is precisely designed so that the impedance is unchanged from one tube to the other, the incident wave is entirely transmitted to the second tube. To devise such matched impedance interfaces can be useful to protect a transducer from a corrosive fluid without inducing poor transmission of the acoustical waves. For example, a specific material called “rho-c rubber” is suitable to cover the face of underwater sound transducers because its specific impedance is nearly equal to that of seawater, see Appendix A2.
252
Fluid-structure interaction
4.2.1.5 Reflected and transmitted waves through three media The problem of sound reflection and transmission through three distinct media is of considerable importance from the practical standpoint. In particular, it concerns the very common occurrence of a fluid separated from another fluid by a thin solid layer. The presentation given here follows closely that made in [BLA 00].
Figure 4.4. Travelling plane waves through three distinct fluid (or solid) columns
Let us consider a pipe made of three straight tube elements of same crosssectional area, but filled with three distinct fluids. As shown in Figure 4.4, the origin is taken at the first interface. The acoustic field in the first two media comprises a forward and a backward component. It is written as: p1 = p1( ) e (
i ω t + k1 x )
+
( + ) i ( ω t − k2 x )
p2 = p2 e
+ p1( ) e (
i ωt − k1 x )
−
p1( ) e (
( − ) i (ωt + k2 x )
+ p2 e
i ωt + k1 x )
+
; q1 = ; q2 =
− p1( ) e ( Z1
p2( ) e (
i ω t − k2 x )
+
i ωt − k1 x )
−
− p2( ) e ( Z2 −
i ω t + k2 x )
[4.23]
where Z1 , Z2 and Z3 are the impedances of the tube elements. Continuity conditions of pressure and volume velocity at the first interface lead to: p1( ) + p1( ) = p2( ) + p2( +
−
+
−)
;
p1( ) − p1( Z1 +
−)
=
p2( ) − p2( Z2 +
−)
[4.24]
The acoustic field within the third column, assumed to extend indefinitely in the forward direction, is: p3 = p3( ) e ( +
i ω t − k3 ( x − L ) )
; q3 =
p3( ) e ( Z3 +
i ω t − k3 ( x − L ) )
Continuity conditions at the second interface lead to:
[4.25]
Plane acoustical waves in pipe systems
p2( ) e − ik2 L + p2( ) e + ik2 L = p3 ; +
−
p2( ) e − ik2 L − p2( ) e + ik2 L p3( ) = Z2 Z3 +
−
253
+
[4.26]
Adding and then subtracting the two equations [4.26], yields: Z ⎞ Z ⎞ + + ⎛ − + ⎛ 2 p2( ) e − ik2 L = p3( ) ⎜ 1 + 2 ⎟ ; 2 p2( ) e + ik2 L = p3( ) ⎜ 1 − 2 ⎟ ⎝ Z3 ⎠ ⎝ Z3 ⎠
[4.27]
Adding the two equations [4.24] yields: 2 p1( ) = +
Z ⎞ Z ⎞ + ⎛ − ⎛ p2( ) ⎜ 1 + 1 ⎟ + p2( ) ⎜ 1 − 1 ⎟ ⎝ Z2 ⎠ ⎝ Z2 ⎠
[4.28]
Then, eliminating the intermediate field variables in medium (2) between [4.27] and [4.28], the following overall transmission coefficient is finally obtained as: T=
p3( p1(
+) +)
=
2 ⎛ Z1 ⎞ ⎛ Z1 Z2 ⎞ ⎜ 1 + ⎟ cos k2 L + i ⎜ + ⎟ sin k2 L ⎝ Z3 ⎠ ⎝ Z2 Z3 ⎠
[4.29]
The primary reflection coefficient can be related to T, by substituting the values of + − the pressure fields p2( ) and p2( ) given by [4.27] into the first equation [4.24]: R=
− ⎛ ⎞ Z p1( ) = T ⎜ cos k2 L + i 2 sin k2 L ⎟ − 1 ⇒ (+) Z3 p1 ⎝ ⎠
⎛ Z1 ⎞ ⎛ Z2 Z1 ⎞ ⎜ 1 − ⎟ cos k2 L + i ⎜ − ⎟ sin k2 L Z3 ⎠ ⎝ Z3 Z2 ⎠ R=⎝ ⎛ Z1 ⎞ ⎛ Z1 Z2 ⎞ ⎜ 1 + ⎟ cos k2 L + i ⎜ + ⎟ sin k2 L Z 3 ⎠ ⎝ ⎝ Z2 Z3 ⎠
[4.30]
Since the acoustic properties of each medium are entirely described by their specific impedance and geometry (length and cross-sectional area), formulas [4.29] and [4.30] hold for solids or fluids, as well. Furthermore, extension to the case of acoustic elements of distinct cross-sectional areas is immediate as it suffices to replace the specific impedance of the medium by the impedance of the tube element. Various special cases are worth discussing in relation to the compressibility parameter k2 L in the intermediate medium. 1. Half wavelength case: k2 L = nπ
254
Fluid-structure interaction
If the length of the intermediate column is an integral number of half wavelengths in the intermediate medium, the relations [4.29] and [4.30] simplify into: 2Z3 ( −1)n Z3 + Z1
T=
; R=
Z3 − Z1 Z3 + Z1
[4.31]
Except for the phase term ( −1) n , the formulas [4.31] are the same as those which hold at the interface between two media, see formulas [4.22]. Therefore, at the specific frequencies f n = nc2 / 2 L , the presence of the intermediate medium is not felt. 2. Quarter wavelength case: k2 L = ( 2n − 1) π / 2 If the length of the intermediate column is an integral number of a quarter wavelength in the intermediate medium, the relations [4.29] and [4.30] simplify into: T=
2Z2Z3
Z1Z3 + (Z2 )
2
e
i
( 2 n −1)π
(Z ) − Z1Z3 R= 2 2 (Z2 ) + Z1Z3 2
;
2
[4.32]
As a particularly interesting case, primary reflection vanishes for the special value Z2 = Z1Z3 of the impedance of the intermediate medium and the special frequencies f n = ( 2n − 1) c2 / 4 L . Furthermore, it is also of interest to verify that the condition of no reflection of the primary wave also correspond to a full transmission of the wave energy from medium (1) to medium (3). Substituting the value Z2 = Z1Z3 into the transmission factor of the pressure wave [4.32], it is found that: T=
Z3 i e Z1
( 2 n −1)π 2
[4.33]
On the other hand, the formulas [3.28] to [3.30] of Chapter 3, are used to calculate the power radiated. The instantaneous power radiated by the wave travelling within the medium (1) is: P1
(+)
(
p1( ) cos (ωt − k1 x ) ( + ) i (ωt − k1 x ) ( + ) i (ωt − k I x ) = ∫∫ I .nd S = Re p1 e Re q1 e = (S f ) Z1
(
) (
)
+
)
2
[4.34]
The mean power radiated per cycle follows as: p( ) ) ( = +
P1
(+)
1
Z1
2
2π / ω
⎛ ω ⎞ ⌠⎮ ⎜ ⎟⎮ ⎝ 2π ⎠ ⎮⌡0
p( ) ) ( ( cos (ωt − k x ) ) dt = 2Z +
2
1
1
1
2
=
( ))
ZI q1( 2
+
2
[4.35]
Plane acoustical waves in pipe systems
255
In the same way, the mean power radiated per cycle within the medium (3) is: p( ) ) ( ( ) P = +
3
+
3
2Z3
2
=
( ))
Z3 q3(
+
2
2
[4.36]
Using the transmission coefficient [4.33], it is found that the wave energy is fully transmitted from medium (1) to medium (3) as should be: 2
+ + P3 ( ) ⎛ Z1 ⎞ ⎛ p3( ) ⎞ = ⎜ ⎟ ⎜ (+) ⎟ = 1 (+) ⎜ ⎟ P1 ⎝ Z3 ⎠ ⎝ p1 ⎠
[4.37]
3. Thin layer approximation: k2 L << 1 In the thin layer approximation, the formulas (4.29) and (4.30) simplify into: ⎛ Z1 ⎞ ⎛ Z2 Z1 ⎞ ⎜ 1 − ⎟ + i ⎜ − ⎟ k2 L Z3 ⎠ ⎝ Z3 Z2 ⎠ 2 T= ; R=⎝ ⎛ Z1 ⎞ ⎛ Z1 Z2 ⎞ ⎛ Z1 ⎞ ⎛ Z1 Z2 ⎞ ⎜ 1 + ⎟ + i ⎜ + ⎟ k2 L ⎜ 1 + ⎟ + i ⎜ + ⎟ k2 L Z Z Z 3 ⎠ 3 ⎠ ⎝ ⎝ 2 ⎝ Z3 ⎠ ⎝ Z2 Z3 ⎠
[4.38]
Further simplifications now depend on the characteristic impedance ratios. As a first case, if k2 L is so small that the imaginary parts in R and T can be neglected, presence of the intermediate medium is not felt, this provided the acoustic wavelengths are large enough. As a first example of practical importance, let us consider a column of water separated from a column of gas by a thin sheet of solid, Perpex for instance. At standard temperature and pressure, the specific impedance of water is about 1.4106 rayls ( ρ f 1000 kg/m 3 c f 1500 m/s ) and that of air is about 400 rayls
( ρ f 1.2 kg/m 3 , c f 344 m/s ). Finally that of Perpex is nearly
2106 rayls . Therefore, the imaginary parts of R and T are less of about one percent than the real parts, as soon as L ≤ 4mm and λ ≥ 1 m .
As a second example, we consider the transmission of sound through an intermediate medium of large impedance: Z2 >> Z1 = Z3 . In that case, T is approximated as: T=
1 1 = ⎛ Z2 k2 L ⎞ ⎛ ωm ⎞ 1+ i ⎜ ⎟ 1+ i ⎜ ⎟ ⎝ 2Z1 ⎠ ⎝ 2Z1 ⎠
[4.39]
where m is the surface density, or mass per unit area, of the solid medium. With the aid of relation [4.37], the transmission loss in terms of radiated power is found to be:
256
Fluid-structure interaction
⎛ (+) + P3 ( ) ⎜ p3 = + P1 ( ) ⎜ p1( + ) ⎝
2
⎞ 1 ⎟ = TT * = 2 ⎟ ⎛ ωm ⎞ ⎠ 1+ ⎜ ⎟ ⎝ 2Z1 ⎠
[4.40]
The sound attenuation through the solid barrier is thus found to be negligible in the low frequency range such that ω << ρ 2 L / ρ1c1 and proportional to (ω m ) in the 2
high frequency range ω >> ρ 2 L / ρ1c1 , which is the so-called mass law of sound barriers at normal incidence, further described in Chapter 7 subsection 7.2.2. 4.2.1.6 Boundary conditions and terminal impedances
Figure 4.5. Uniform tube of finite length
Let us consider now a tube of finite length L, see Figure 4.5. The ends may be named in accordance with the positive direction along the tube axis as the inlet, at x = 0, and as the outlet, at x = L. The homogeneous and linear boundary conditions to be satisfied at the tube inlet and outlet are of the general type:
[ ap + bq]in ,out = 0
[4.41]
where the coefficients a and b can depend on ω. Once more, the acoustic properties of these terminations can be characterized by an impedance and the conditions of sound propagation through the tube end is the same as if the real tube was connected to a fictitious tube of impedance Z2 = Z ( L ) . For instance, if the tube is terminated by a pressure node at the outlet Z ( L ) = 0 , using the relations [4.22] it is found that the pressure wave incident on the outlet is fully reflected with a sign change ( R = −1; T = 0 ). If the waves are fully reflected at the tube ends, the interferences between the incident and reflected waves give rise to a system of standing waves, written as:
Plane acoustical waves in pipe systems
p ( x; k ) = A sin kx + B cos kx
q ( x; k ) =
iS f ρ f cf
257
[4.42]
( A cos kx − B sin kx )
[4.43]
where the wave number k is assumed to be real. The inlet impedance is: p q
Zin =
= −i x =0
ρ f cf B Sf A
[4.44]
Going a step further, it is found convenient to define the dimensionless inlet impedance αin as follows: −i ρ f c f B = tanα in ⇒ Zin = tanα in = −iZ tanα in A Sf
[4.45]
In a similar way, the outlet dimensionless impedance is defined as: Zout =
p q
= x=L
+i ρ f c f Sf
tan α out = +iZ tan α out
[4.46]
where Z is the impedance of the tube. As a convention, the sign is changed when passing from the inlet to the outlet impedance to mark that at the inlet volume velocity q stands for a flow entering into the tube, whereas at the outlet it stands for a flow leaving the tube. As a first interesting application, let us consider the reflection and transmission laws at the tube terminations, which are implied by the inlet and outlet impedances as defined by the relations [4.45] and [4.46]. According to the formulas [4.22], the coefficient of reflection at the tube outlet is: Rout =
Zout − Z+ i tan α out − 1 = = −e −2iα Zout + Z+ i tan α out + 1
out
[4.47]
The same result as [4.47] holds at the inlet, provided one takes into account that the incident wave travels backward. Accordingly, the coefficient of reflection at the tube inlet is: Rin =
Zin − Z− −i tan α in + 1 =− = −e −2iαin Zin + Z− i tan α in + 1
[4.48]
For full reflection and zero transmission of the waves to occur at the tube terminations, the necessary and sufficient condition is that the modulus of the reflection coefficient is unity. Hence, from the results [4.47] and [4.48] it can be immediately inferred that, depending whether a dimensionless impedance is a real or a complex number, its nature is conservative or not.
258
Fluid-structure interaction
On the other hand, substituting the standing waves [4.42] and [4.43] into the outlet impedance [4.46] implies: p q
=
iρ f c f
x=L
Sf
tan αout = −
iρ f c f Sf
tan ( kL + αin )
[4.49]
Equation [4.49] stands for a characteristic equation which relates the circular frequency of the wave to the impedances at the ends of the tube: kL =
ωL = ϖ = nπ − (α in + α out ) ; n = 0,1, 2,... cf
[4.50]
As further illustrated just below, the relations [4.42] and [4.43] describe standing waves so long as k and ω are real, which implies that the dimensionless impedances are real valued, as already pointed out just above. 1. Pressure node If the tube communicates with the free atmosphere, provided surface tension, gravity, and more important local 3D effects are negligible, the appropriate boundary conditions read as: p
x =0
= 0 ⇔ αin = nπ ; n = 0,1, 2,...
p
x=L
= 0 ⇔ αout = nπ ; n = 0,1, 2,...
[4.51]
At the inlet, the backward wave is completely reflected as a forward wave, whereas the reverse occurs at the outlet. In both cases the incident and reflected waves are related by the following reflection law: p+ = − p− q+ = + q−
[4.52]
As there is no wave transmitted outside, acoustic energy is conserved. 2. Volume velocity node If the tube is closed by a rigid and fixed wall, the appropriate boundary conditions read as: π + nπ ; n = 0,1, 2,... 2 π = + nπ ; n = 0,1, 2,... 2
q
x =0
= 0 ⇔ αin =
q
x=L
= 0 ⇔ αout
The incident and reflected waves are related by the following reflection law:
[4.53]
Plane acoustical waves in pipe systems
p+ = + p−
259
[4.54]
q+ = − q−
3. Elastic impedance Elastic impedance can be defined to model a piston connected to a spring, or even a flexible wall, provided the structural mass can be neglected. Denoting Kin the stiffness coefficient of the spring acting at the inlet, the condition of mechanical equilibrium at the inlet is written as: Kin X in = − pin S f ⇔ K in
qin K = − pin S 2f ⇔ Zin = − in 2 iω iω S f
Kout X out = + pout S f ⇔ Kout
qout K = + pout S 2f ⇔ Zout = + out2 iω iω S f
[4.55]
Thus, by using the relations of definition [4.45] and [4.46]: −
−i ρ f c f Kin = tan α in 2 iω S f Sf
; +
+i ρ f c f Kout = tan α out 2 iω S f Sf
[4.56]
The dimensionless impedances can be written as: ⎛ Kin α in = − tan −1 ⎜ ⎜ ωρ c S f f f ⎝
⎞ ⎛ Kin ⎞ −1 ⎟⎟ = − tan ⎜⎜ ⎟⎟ ⎠ ⎝ K f (λ ) ⎠
⎛ Kout = − tan −1 ⎜ ⎜ ωρ c S f f f ⎝
⎞ ⎛ Kout ⎞ −1 ⎟⎟ = − tan ⎜⎜ ⎟⎟ ⎠ ⎝ K f (λ ) ⎠
α out
[4.57]
where K f ( λ ) is a stiffness coefficient which characterizes the fluid column elasticity for a plane wave of circular frequency, ω or wavelength λ: K f ( λ ) = ωρ f c f S f =
2π E f S f ω ρ f c 2f S f = λ cf
[4.58]
As could be expected, for a given harmonic wave, depending on whether the stiffness coefficient of the piston tends to zero or to infinity, the condition of a pressure node (volume velocity antinode) or that of a volume velocity node (pressure antinode) is recovered. On the other hand, so long as the fluid is compressible, for a given value of the structural stiffness coefficient, depending on whether the frequency tends to zero or to infinity, the condition of a volume velocity node, or that of a pressure node is recovered. In other terms, in the low frequency range, the solid tends to be stiffer than the fluid and in the high frequency range the
260
Fluid-structure interaction
reverse occurs. The reflection law for the pressure and volume velocity waves can be obtained by using the coefficients [4.47] and [4.48]. 4. Inertial impedance The concept of inertial impedance has been already introduced in Chapter 2 subsection 2.2.2.4 in relation to the problem of modelling small orifices. Inertia of the fluid which oscillates through the orifice can be accounted for by using the impedance [2.38] repeated here as: Zin = Zout =
−iω ρ f sf +iω ρ f sf
= =
−i ρ f c f Sf +i ρ f c f Sf
⎛ ω S f tan αin ⇒ αin = tan −1 ⎜ ⎜c s ⎝ f f tan αout ⇒ αout
⎞ ⎟⎟ ⎠
⎛ ω S f = tan ⎜ ⎜c s ⎝ f f −1
⎞ ⎟⎟ ⎠
[4.59]
With the aid of formula [2.34], where h is substituted by , the impedance of the orifice can be further expressed as: ω S f sf cf
=
ω L 8 ⎛ D ⎞ ⎛ D ⎞ 16 ⎛ L ⎞ ⎛ D ⎞ ⎛ D ⎞ ⎜ ⎟⎜ ⎟ = ⎜ ⎟⎜ ⎟⎜ ⎟ c f 3π ⎝ d ⎠ ⎝ L ⎠ 3 ⎝ λ ⎠ ⎝ d ⎠ ⎝ L ⎠
[4.60]
where D is the diameter of the tube, d that of the orifice, L the length of the tube and λ the wavelength. More generally, masses M in and M out at the tube inlet and outlet respectively can be modelled as the impedances: Zin = Zout
ω 2 M in X s −iω M in −i ρ f c f = = tan αin S ( iω SX s ) S 2f Sf
−ω 2 M out X s +iω M out +i ρ f c f = = = tan αout S f ( iω SX s ) S 2f Sf
[4.61]
Whence the dimensionless impedances: ⎛ ω M in αin = tan −1 ⎜ ⎜ρ c S ⎝ f f f
⎞ ⎛ M in ⎞ −1 ⎟⎟ = tan ⎜⎜ ⎟⎟ ⎠ ⎝ M f (λ ) ⎠
⎛ ω M out = tan ⎜ ⎜ρ c S ⎝ f f f
⎞ ⎛ M out ⎞ −1 ⎟⎟ = tan ⎜⎜ ⎟⎟ ⎠ ⎝ M f (λ ) ⎠
αout
−1
[4.62]
where M f ( λ ) is a mass coefficient which characterizes the fluid column inertia for a plane wave of circular frequency ω or wavelength λ:
Plane acoustical waves in pipe systems
M f (λ ) =
ρf Sfλ
261
[4.63]
2π
As could be expected, at a given frequency, depending whether the mass coefficient of the piston tends to zero or to infinity, the condition of a pressure node or that of a pressure antinode is recovered. On the other hand, for a given structural mass coefficient, depending whether the frequency tends to zero or to infinity, the condition of a pressure node or that of a pressure antinode is recovered. In other terms, in the low frequency range, the solid inertia tends to be smaller than that of the fluid and in the high frequency range the reverse occurs. Finally, it may be noticed that in formula [4.39], the thin layer approximation of the three medium problem leads to treatment of the intermediate medium as the inertial impedance Z2 = iωρ f L / S f . 4.2.1.7 Radiation damping and complex impedance Independently from the conservative or dissipative nature of the boundary conditions, the harmonic waves inside the tube can be described, formally at least, either as travelling waves or as stationary waves. A few elementary manipulations lead to the following relationships between the two sets of coefficients of equations [4.13] and [4.42]: 1 1 ( B + iA) ; A− = ( B − iA) 2 2 A = −i ( A+ − A− ) ; B = A+ + A− A+ =
[4.64]
As an immediate consequence, it is found that a travelling forward wave verifies: B = +iA
[4.65]
whereas a travelling backward wave verifies: B = −iA
[4.66]
Substituting the relation [4.65] into equations [4.45] and [4.46], we obtain: tan αin = +i ⇔ Zin =
+ρ f c f Sf
; tan α out = −i ⇔ Zout =
+ρ f c f Sf
[4.67]
As expected, the expression [4.16] of the tube impedance for a travelling forward wave is recovered. In a similar way, substituting the relation [4.66] into the equations [4.45] and [4.46], the characteristic tube impedance for a travelling backward wave is recovered: tan αin = −i ⇔ Zin =
−ρ f c f Sf
; tan α out = +i ⇔ Zout =
−ρ f c f Sf
[4.68]
262
Fluid-structure interaction
Recalling the trigonometric formula, tan ( a ± ib ) =
sin 2a ± i sinh 2b cos 2a + cosh 2b
[4.69]
it follows that: tan α = ±i ⇔ α = ±i∞
[4.70]
To interpret physically such impedances, let us assume that the tube is terminated at the outlet by the impedance αout = −i∞ . Substituting this value into the reflection coefficient [4.47] it is immediately verified that no wave is reflected. On the other hand, since αout = −i∞ corresponds to the characteristic impedance of the tube for a forward wave, the latter is fully transmitted through the outlet, in agreement with the transmission coefficient [4.22], where Z2 = Z1 = + ρ f c f / S f . As a consequence, the acoustic energy of the outgoing wave is never returned to the tube. The process of energy loss through wave radiation in an infinite medium is known as radiation damping. In the same way, if the tube is terminated at the inlet by the impedance αin = −i∞ , an incident backward wave is not reflected but fully transmitted through the inlet as an outgoing wave. Hereafter, presentation will be restricted to real dimensionless impedances up to Chapter 7 which is devoted to the study of dissipative processes, including radiation damping. 4.2.1.8 Acoustical modes in a uniform tube The uniform tube of Figure 4.5 is assumed to be terminated by the real dimensionless impedances αin and αout in such a way that interference between incident and reflected waves give rise to standing waves, called acoustical modes, or resonances. Starting from the equation [4.12], the natural modes in the plane wave approximation, often called pipe modes, are the non trivial solutions of the modal system: 2 ⎧ d2p ⎛ ω ⎞ ⎪ +⎜ ⎟ p =0 dx 2 ⎜⎝ c f ⎟⎠ ⎨ ⎪ ⎩α in and α out known functions of ω.
[4.71]
Of course the system [4.71] is the same as that which governs the longitudinal natural modes of vibration of a straight and uniform beam. Starting from the general solution [4.42], and the inlet impedance [4.44], we obtain: ⎛ ωx B ωx + cos p( x; ω ) = A ⎜ sin ⎜ cf A cf ⎝
where C = A / cos α in .
⎞ ⎛ ωx ⎞ + αin ⎟ ⎟⎟ = C sin ⎜⎜ ⎟ ⎠ ⎝ cf ⎠
[4.72]
Plane acoustical waves in pipe systems
263
The relations [4.49] and [4.50], which connect the pulsation to the impedances at the tube inlet and outlet, are used to determine the eigenfrequencies: ⎛ ωL αout = − ⎜ + α in ⎜c ⎝ f
⎞ ωL ⎟⎟ + nπ ⇒ n = nπ − (α in + α out ) ; n = 1, 2,.. cf ⎠
[4.73]
Whence the natural frequencies and mode shapes normalized to a pressure antinode of unit magnitude: fn =
c f ( nπ − (α in + α out ) )
2π L ⎛ x ( nπ − (α in + α out ) ) p ϕ n( ) ( x ) = sin ⎜ + αin ⎜ L ⎝
⎛ x ( nπ − (α in + αout ) ) X + αin ϕ n( ) ( x ) = cos ⎜ ⎜ L ⎝
⎞ ⎟ ⎟ ⎠
[4.74]
⎞ ⎟ ⎟ ⎠
The above calculation illustrates the pertinence of relations [4.45] and [4.46] to define the dimensionless impedances since the trigonometric algebra is made very easy leading to the compact analytical expressions [4.74]. Two distinct mode shapes p X ϕ n( ) and ϕ n( ) can be defined, according to whether the field variable selected is the pressure or the displacement. They comply with the norm condition of unit p maximum magnitude. Since ϕ n( ) is obtained here by solving the modal problem [4.71], ϕ n( ) is deduced afterwards by using either the mass equation ([4.9] or the momentum equation [4.10]. It is of interest to discuss shortly a few particular cases of common application. X
1. Tube terminated by two pressure nodes ( αin = αout = nπ ) In practice, this case corresponds to the idealized case of a tube open at each ends. The so-called half wavelength resonances are obtained: nπ x L nπ x X ϕ n( ) ( x ) = cos L ϕ n(
p)
( x ) = sin
;
fn =
;
fn =
nc f 2L nc f 2L
; n = 1, 2,...
[4.75] ; n = 0,1, 2,...
It can be noticed that to recover the free rigid mode of longitudinal translation, the mode shapes must be expressed in terms of the displacement variable. 2. Tube terminated by two volume velocity nodes ( αin = α out = ( 2n + 1) π / 2 )
264
Fluid-structure interaction
In practice, this case corresponds to a tube stopped at both ends by rigid and fixed walls. Again half wavelength resonances are obtained: (n − 1)c f (n − 1)π x ; fn = ; n = 1, 2,... L 2L nc nπ x X ; f n = f ; n = 1, 2,... ϕ n( ) ( x ) = sin 2L L ϕ n(
p)
( x ) = cos
[4.76]
The mode at zero frequency and constant pressure is the counterpart, in terms of stress, of the free rigid mode which occurs in terms of displacement. 3. Tube terminated by a pressure node at one end and by a volume velocity node at the other If the pressure node is located at the inlet ( α in = 0, αout = π / 2 ), the quarter wavelength resonances are found: fn = ( p)
ϕn
(2n − 1)c f 4L
; n = 1, 2,...
πx ( x ) = sin ⎛⎜ ( 2n − 1) ⎞⎟ 2L ⎠ ⎝
;
(X )
ϕn
πx ( x ) = cos ⎛⎜ ( 2n − 1) ⎞⎟ 2L ⎠ ⎝
[4.77]
If the pressure node is located at the outlet ( αin = π / 2, αout = 0 ), the natural frequencies remain obviously the same while the pressure and displacement mode shapes are exchanged: ϕ n(
p)
( x ) = cos ⎛⎜ ( 2n − 1) ⎝
πx⎞ ⎟ 2L ⎠
;
ϕ n(
X)
( x ) = sin ⎛⎜ ( 2n − 1) ⎝
πx⎞ ⎟ 2L ⎠
[4.78]
In both cases, the modes correspond to an odd number of quarter wavelengths. 4. Tube terminated by two identical elastic impedances Substituting the results [4.55] into equation [4.73], the following transcendental equation is found: ωn −
c f ⎛ −1 ⎛ Kin ⎜ tan ⎜⎜ L ⎜⎝ ω ρ ⎝ n f cf Sf
⎞ ⎛ Kout −1 ⎟⎟ + tan ⎜⎜ ⎠ ⎝ ωn ρ f c f S f
⎞ ⎞ nπ c f ; n = 1, 2,... ⎟⎟ ⎟⎟ = L ⎠⎠
which can be conveniently discussed using a numerical example.
[4.79]
Plane acoustical waves in pipe systems
265
Figure 4.6. Characteristic equation of the tube terminated by elastic impedances
Let us consider a circular cylindrical tube (length 1 m, internal radius 1 cm) filled with air at STP. Adopting the coefficients K in = K out = K = 400ρ f c f S f , the results of the modal analysis restricted to the first two modes are shown graphically in Figures 4.6 and 4.7. Figure 4.6 displays the characteristic equation [4.79]. The horizontal dashed lines correspond to the natural frequencies of the second and the third modes of the tube closed at both ends ( Kin and Kout tending to infinity). The star and the cross indicate the location of the roots of the equation for that asymptotic case, which reduces to: ωn −
c f kπ L
=
nπ c f L
The first root f1( (∞ ) 2
∞)
; n = 0,1, 2,...; k = ±1, ±2,...
= 0 Hz correspond to k = -n. The second and the third roots
(∞ ) 3
f = 170 Hz , f = 340 Hz correspond to k = n - 1 and k = n - 2. As the stiffness decreases, the natural frequencies are found to decrease, except of course that of the K constant pressure mode. In the case of Figure 4.6, f 2( ) 116 Hz and f 3(
K)
= 318 Hz . Figures 4.7 displays the corresponding mode shapes.
Finally, it is also of interest to discuss the asymptotic case of a small stiffness coefficient such that: ⎛ K tan −1 ⎜ ⎜ω ρ c S ⎝ 1 f f f
⎞ K K ⎛ L = ⎟⎟ ⎜⎜ ⎠ ω1ρ f c f S f ρ f S f L ⎝ ω1c f
⎞ ⎟⎟ << 1 ⎠
[4.80]
266
Fluid-structure interaction
Figure 4.7. Pressure mode shapes of the tube terminated by elastic impedances
Substituting the approximation [4.80] into the characteristic equation [4.79], the lowest natural frequency is given by: ω1 =
cf ⎛ 2K ⎜⎜ L ⎝ ω1 ρ f c f S f
⎞ ⎟⎟ ⇒ ω1 = ⎠
2K ρf Sf L
[4.81]
Plane acoustical waves in pipe systems
267
The result [4.81] corresponds to the rigid mode of the fluid column supported by the springs. Such a remark can be further reinforced by substituting the value of ω1 as given by [4.81] into [4.80], which allows one to verify that for this low frequency mode, fluid compressibility is negligible: ω1 L << 1 cf
[4.82]
4.2.1.9 Application to wind musical instruments The basic principle of wind musical instruments like pipe organs, woodwind and brass instruments, is to produce sounds by letting vibrate an air column inside a pipe. The sound is generated by coupling the air column, which acts as a resonator, to a flow-control mechanism that converts steady wind supply into oscillations of the air column. Of the acoustical power produced inside the pipe, only a small percentage is radiated by the openings of the instrument and made available to the listener as further described in Chapter 7. The sounds produced must meet several criteria, some measurable and others not, to be perceived as musically pleasant sounds. The pitch is an attribute of auditory sensation which is primarily dependent upon the frequency of the sound. Though the average ear can distinguish 1400 discrete frequencies, only 120 tones are retained in the so-called equally tempered scale covering the hearing range of 16 to 16000 Hz, see for instance [RAY 94], [OLS 67]. At this point, a nutshell of background information may be appropriate. In the equally-tempered scale, each frequency doubling is divided in twelve notes with constant frequency ratios r , so that 2 f i / f i = r12 . For historical, musical and ergonomic reasons, in keyboard instruments these twelve notes consist on seven white keys and five black keys (in groups of two and three), the basic musical interval between every two consecutive notes being r = 21/12 = 1.0595 , which is called a “half-tone”. Frequency-doubling can be obtained going through thirteen consecutive keys (12+1) – which is called a chromatic scale, entirely composed of half-tones. However, a large body of classical and popular music is anchored on uneven scales with fewer notes, consisting on half-tones and multiples of the halftone – the “tone” r 2 = 1.1225 being the most common interval. Among those scales, the basic one obtained by playing the seven consecutive white keys – called a diatonic scale – is of paramount significance in western music. In this scale, frequency doubling is obtained going through height notes (7+1) – hence the commonly used term “octave” – consisting on five tone intervals and two half-tones. As most music is shaped from these musical intervals, they can obviously be perceived by any person. A smaller interval such as the “quarter-tone” r1/ 2 = 1.0293 is easily perceived by musicians, as well as by many untrained people. Intervals smaller than the “eight-tone” r1/ 4 = 1.0145 are often not perceived as different frequencies.
268
Fluid-structure interaction
Furthermore, the musical tones are generally complex tones superposing a fundamental, which stands for the lowest frequency component of the complex tone and several harmonics of the fundamental. In other terms, the pressure signal corresponding to the complex tone may be represented as a Fourier series. Accordingly, the frequency of the signal, equal to that of the fundamental specifies the pitch of the tone while the relative amplitude and phasing of the harmonics specifies the timbre. Linear superposition of simple tones (that is harmonic sounds) whose frequencies are not related as in a Fourier series are likely to produce unpleasant sounds, including in particular beats between adjacent components. In this respect, wind instruments using a column of air of constant cross sectional area as a resonator, are very convenient and can be understood as the fluid counterpart of string instruments. In a string instrument, the frequency can be varied continuously by pressing the finger at any place along the string to prescribe a node of displacement, which is equivalent to changing the effective length of the vibrating string. In relatively few wind instruments, like the trombone, the length of the air column can also be varied continuously. In most instruments, the acoustical length is varied by closing or opening a few holes judiciously located along the pipe to adjust properly the tone pitch to the musical scale. When open, a hole acts practically as a pressure node. Actually, as shown later in Chapter 7, the acoustical dimensionless impedance of a hole is not exactly zero, but a complex number, the real part of which accounts for the inertia of the fluid vibrating back and forth through the hole and the imaginary part for the wave radiated outside the pipe. On the other hand, if the pipe is open at both ends, it can be assumed to be terminated nearly by two pressure nodes. So, the wavelength of the fundamental is twice the tube length and all the harmonics are multiple integers of the fundamental, see formula [4.75]. The open flue pipes of organs and the flute are two typical instruments behaving nearly like a circular cylindrical pipe open at both ends. The sound is produced by an air jet, cf. subsection 4.3.1.1. The jump to what musicians call the next higher pitched octave is possible since the frequency of the second mode is double that of the fundamental. This is not possible if the tube is closed at one end and open at the other, as the fundamental corresponds now to a quarter wavelength and only the odd harmonics are produced, see formula [4.77]. Since a stopped pipe sounds an octave below the pitch of a similar pipe open at both ends, there is an economical advantage in using stopped tubes especially in the low register requiring tube lengths of several meters. The interest of using a stopped pipe instead of an open pipe is not only economical but is also musical as it yields a very distinct timbre from an open one of the same fundamental frequency. The stopped flue pipes of organs and the clarinet are two typical instruments behaving nearly like a circular cylindrical pipe open at one end and closed at the other. In the case of the organ, the pipe is closed at the outlet and the sound is produced by an air jet at the inlet. In the case of the clarinet, the pipe is open at the outlet and nearly closed at the inlet where the sound is produced by letting vibrate a mechanical reed which consists of a flat piece of cane. Instead of the jump to the next octave which is obviously impossible, the jump to the next 12th (one octave plus a fifth) is available. As discussed in the
Plane acoustical waves in pipe systems
269
next subsection, wind instruments also make use of conical tubes, for instance the oboe family and the saxophones. 4.2.1.10 Horns: Webster and Schrödinger equations The wave equation [4.11], particularized to the case of a homogeneous fluid, is written here as: d ⎛ dp ⎞ 2 d 2 p S f ′ dp + k2 p = 0 ⎜Sf ⎟ + k Sf p = 0 ⇔ 2 + dx ⎝ dx ⎠ dx S f dx
[4.83]
Equation [4.83] is broadly called the Webster equation by the acousticians. The horns are a particular case such that the derivative S f ′ = dS f / dx is positive. Therefore, the characteristic tube impedance is smaller at the outlet than at the inlet. Horns are of common use to give a preferential direction to the emitted sound, as further explained in Chapter 5 subsection 5.3.4.8 and as an impedance-matching device to increase the radiated acoustic power output of a source. The acoustical source is located at the inlet, that is where the impedance is high and power is radiated from the outlet where the impedance is low, as clarified in Chapter 7 subsection 7.2.1.4. An interesting change of variable exists which allows one to transform the wave equation [4.83] into the famous Schrödinger equation which can be considered as the cornerstone of quantum mechanics. Assuming circular cross-sections of radius R(x) for convenience, the new variable Ψ is defined as: Ψ = p S f = π pR ⇒ dp 1 ⎛Ψ ′ Ψ R ′ ⎞ = − 2 ⎟ ⎜ dx R ⎠ π ⎝ R 2 d p 1 ⎛Ψ ′′ Ψ ′R ′ Ψ R ′′ Ψ R ′2 ⎞ = −2 2 − 2 +2 3 ⎟ ⎜ 2 dx R R R ⎠ π ⎝ R
[4.84]
Once more the prime stands for a derivation with respect to x. Substituting the expressions [4.84] into [4.83], the following wave equation is obtained: d 2Ψ + ( k 2 − F )Ψ = 0 dx 2
[4.85]
where F(x) is termed the horn function defined as: F ( x) =
R ′′ χ xx χθθ R
[4.86]
As indicated in [4.86], if the variation rate of R is small enough, the second derivative R ′′ is a reasonable approximation of the curvature of the wall generatrix
270
Fluid-structure interaction
and F(x) may be expressed as the product of the transverse χθθ 1/ R and longitudinal ( χ xx R′′ ) curvatures, that is essentially the total, or Gaussian, curvature of the pipe wall, as already defined in [AXI 05], Chapter 7. The general solution of [4.85] is written as:
(
Ψ ( x; k ) eiωt = eiωt Ψ + e
− ix k 2 − F ( x )
+Ψ − e
+ ix k 2 − F ( x )
)
[4.87]
Whence the pressure wave: p ( x; k ) eiωt =
(
eiωt − ix Ψ +e Sf
k2 −F (x)
+Ψ − e
+ ix k 2 − F ( x )
)
[4.88]
The important and immediately apparent point is that depending on whether k 2 − F ( x ) is positive, or negative, the solution [4.88] stands for undamped travelling waves, or for spatially evanescent waves, whose amplitude decreases in an exponential way with the distance from the source. The cut-off for passing from evanescent to travelling wave solutions may be expressed either in terms of wave number, or frequency as: ⎧⎪ 0 if F ( x ) ≤ 0 kc ( x ) = ⎨ ⎪⎩ F ( x ) if F ( x ) > 0
0 if F ( x ) ≤ 0 ⎧ ⎪ fc ( x ) = ⎨ c F ( x ) f if F ( x ) > 0 ⎪ ⎩ 2π
[4.89]
Results of this nature have a counterpart of considerable importance in quantum mechanics. Though the subject can be rightly considered as of peripheral interest in the framework of the present book, it is worthwhile to make a short presentation. As a basic principle of quantum mechanics, the presence of a particle at some place and at some time can be asserted in terms of probability only. For that purpose, a matter wave is defined, denoted Ψ . Considering the simplest case of a one-dimensional space, the probability for a particle to lay in the interval x, x+dx is ΨΨ *dx . Ψ is governed by the Schrödinger equation, which can be derived by analogy with the pressure wave equation and using the de Broglie relationship between the matter wave number and energy of the particle, which is initially expressed as: kp =
2π 2π p = λp
[4.90]
where k p denotes the wave number and λ p the wavelength of the matter wave. Here, p stands for the linear momentum of the particle, instead of the usual
Plane acoustical waves in pipe systems
271
notation p to avoid confusion with pressure. Finally, denotes the Planck constant. The kinetic energy of the particle is: Eκ =
p2 2m
[4.91]
where m is the mass of the particle at rest. The total energy is written as the sum of the kinetic and potential energy: ET = Eκ + Ep
[4.92]
The momentum of the particle is thus written as: p=
2m (ET − Ep )
[4.93]
By analogy with the wave equation of an elastic medium, the one-dimensional Schrödinger equation is found to be: d 2Ψ d 2Ψ 8π 2 m 2 + k Ψ = 0 ⇔ + 2 (Eκ − Ep )Ψ = 0 p dx 2 dx 2
[4.94]
Hence, if Ep is greater than Eκ , the particle remains trapped in the potential sink. However, the probability to find the particle outside the sink is not zero but vanishes exponentially with the distance. This is known as the tunnelling effect which has found many applications, in electronics in particular. Returning to the subject of sound waves in horns, according to the second relation [4.89], the threshold above which the travelling wave solution holds is found to vary with the position along the horn, except if the horn function is constant. By prescribing the condition of a constant horn function, hence a single cut-off value, a particular family of horns is defined, known as the Salmon horns, whose geometry is given by: R ′′ − κ 2 R = 0 ⇒ R ( x ) = R1eκ x + R2 e −κ x = R0 ( cosh κ x + β sinh κ x )
[4.95]
A few particular cases are worth mentioning. First, the exponential horn is such that β = 1 . Then, the case β = 0 defines the catenoidal horns, which have the nice geometrical feature of fitting smoothly in the continuation of a circular cylindrical tube to build a compound horn. Finally, the conical horn arises as a limit case where κ tends to zero while β = 1/ κ x0 . The vertex of the cone is at − x0 . It may be immediately checked that in the conical case, the profile [4.95] becomes: ⎛ x⎞ R ( x ) = R0 ⎜ 1 + ⎟ x0 ⎠ ⎝
[4.96]
272
Fluid-structure interaction
4.2.1.11 Bessel horns Another family of horns often discussed in the specialized literature of musical acoustics is that of the Bessel horns. Starting once more from the wave equation [4.85], let us assume that the bore radius varies as the power law: α
⎛ x⎞ R ( x ) = R0 ⎜ ⎟ ⎝ x0 ⎠
[4.97]
where α is a real dimensionless number. Substituting [4.97] into equations [4.86] and then [4.85], one obtains the differential equation: d 2Ψ ⎛ 2 α (α − 1) ⎞ +⎜k − ⎟Ψ = 0 dx 2 ⎝ x2 ⎠
[4.98]
It turns out that equation [4.98], often called the spherical Bessel equation, can be transformed into an exact Bessel equation by using the following change of variable: Ψ = φ xν ⇒ Ψ ′ = xν φ ′ + ν xν −1φ ; Ψ ′′ = xν φ ′′ + 2ν xν −1φ ′ + ν (ν − 1) xν − 2φ
[4.99]
By substituting the relations [4.99] into [4.98] and performing a few straightforward manipulations, we are led to select ν = 1/ 2 and the following Bessel equation arises: 2 d 2φ 1 d φ ⎛ 2 (α − 1/ 2 ) ⎞ k + + − ⎜ ⎟φ = 0 ⎟ dx 2 x dx ⎜⎝ x2 ⎠
[4.100]
Whence the general solution of [4.98]: Ψ ( x; k ) = x ( AJ α −1/ 2 ( kx ) + BYα −1/ 2 ( kx ) ) = Ajα −1/ 2 ( kx ) + Byα −1/ 2 ( kx )
[4.101]
where A and B are two constants of integration and jα , yα are the spherical Bessel functions of order α and of the first and the second kind respectively (on spherical functions see also Appendix 5). The corresponding pressure wave reads as: p ( x; k ) =
α
x0
R0 x (
α −1/ 2 )
( aJ
α −1/ 2
( kx ) + bYα −1/ 2 ( kx ) )
[4.102]
where a and b are other constants. The conical bore is recovered as the particular case α = 1 with the aid of the identities:
Plane acoustical waves in pipe systems
J 1/ 2 ( kx ) =
2 2 sin kx ; Y1/ 2 ( kx ) = cos kx π kx π kx
273
[4.103]
As shown in this subsection, the acoustical modes of a conical tube present interesting peculiarities worth describing here. Using the equations [4.102] and [4.103], the harmonic pressure field in the conical tube of Figure 4.8 is written as: p ( x; k ) =
x0 ( A sin kx + B cos kx ) 0 ≤ x ≤ L x0 + x
[4.104]
Considering first a tube open at both ends, it is immediately noticed that the natural frequencies are the same as those of the tube of same length and constant crosssectional area. Of course, this remarkable property is very attractive so far as the design of a musical instrument is concerned. The pressure mode shapes are written as: ϕn ( x ) =
x0 ⎛ nπ x ⎞ R ( 0 ) ⎛ nπ x ⎞ sin ⎜ sin ⎜ ⎟= ⎟ n = 1, 2,... ; 0 ≤ x ≤ L x0 + x ⎝ L ⎠ R ( x) ⎝ L ⎠
[4.105]
Figure 4.8. Conical bore (an oboe for instance)
The recorder can be mentioned as a family of wind instruments making use of a conical tube open at both ends as a resonator. The smaller radius is located at the outlet, and the taper angle (conical semi-angle) is between 0.5 to 1°. Concerning the musical and practical advantages of such a design the reader is referred to [FLE 98]. Then, we consider the case of a conical tube stopped at the inlet and open at the outlet. A straightforward calculation gives: dp dx
B kL = kx0 = β A −1 x =0 B p x = L = 0 ⇒ = − tan kL A =0⇒
[4.106]
274
Fluid-structure interaction
where β = RL / R0 in agreement with the notation used in [4.95]. Thus, the equation giving the natural frequencies reads as: tan ϖ n =
ϖn ωL where ϖ n = kn L = n cf 1− β
[4.107]
The pressure mode shapes follow as: ϕ n(
p)
( x) =
1 {( β − 1) sin kn x + ϖ n cos kn x} 1 + ( β − 1)( x / L )
[4.108]
Figure 4.9. Natural frequencies versus the flare parameter β of the conical bore (open-open case: dashedt line, stopped-open case: full line)
A priori, the natural frequencies differ from a harmonic sequence. However, provided the cone angle is sufficiently large, they nearly match the values of the circular cylindrical tube open at both ends, so the jump to the next higher pitched octave is possible. Hence, both conical and cylindrical bores are suitable as a resonator of wind instruments, even in the stopped free configuration, as it is in the case of the oboes. Depending on the geometry and boundary conditions, the excitation mechanism and the timbre of the instrument are highly modified. Figure 4.9 displays the first few resonances of a conical tube, L = 50 cm, c f = 340 ms-1 , as plotted versus the flare parameter b. The dashed lines refer to a tube open at both ends and the full lines to a tube stopped at the inlet (radius R0 ) and open at the outlet (radius RL ). The first three pressure
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275
mode shapes are shown in Figures 4.10 to 4.15. The same mode is shown in each individual figure for three geometries β =0.1, 0, 10. Figures 4.10 to 4.12 refer to the open-open case and Figures 4.13 to 4.15 to the stopped-open case.
Figure 4.10. Bore open at both ends: first mode
Figure 4.11. Bore open at both ends: second mode
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Fluid-structure interaction
Figure 4.12. Bore open at both ends: third mode
Figure 4.13. Bore stopped-open: first mode
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277
Figure 4.14. Bore stopped-open: second mode
Figure 4.15. Bore stopped-open: third mode
To conclude this subsection, it is worth emphasizing that only the tubes provided with a bore of either constant cross-sectional area or varying as the square of the position have their natural frequencies ordered exactly as a harmonic sequence. This can be conveniently illustrated by considering the Bessel horn α = 1/2 which is intermediate between the circular cylinder α = 0 and the cone α = 1, see Figure 4.16a. The pressure field [4.102] becomes:
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Fluid-structure interaction
p( x; k ) = AJ 0 ( kx ) + BY0 ( kx ) ; x0 ≤ x ≤ x0 + L
[4.109]
(a)
(b)
(c)
(d) Figure 4.16. (a): Horn profile with α = 0.5 , (b): Characteristic function ( α = 0.5 ), (c): Horn profile with α = 1.5 , (d): Characteristic function ( α = 1.5 )
Assuming for instance that the tube is terminated by two pressure nodes, the natural pulsations are found to be the roots of the following equation: J 0 ( kn x0 ) Y0 ( kn ( x0 + L ) ) − Y0 ( kn x0 ) J 0 ( kn ( x0 + L ) ) = 0
[4.110]
Figure 4.16b plots the function [4.110] versus the pulsation ω , for a Bessel horn with parameters α = 1/2, L = 75 cm, x0 = 5 cm and R0 = 0.5 cm. The fundamental is at about 212 Hz and the next nine partials are at about 442, 671, 899, 1127, 1354,
Plane acoustical waves in pipe systems
279
1582, 1809, 2036 and 2263 Hz. Instead of the pure harmonic series, these frequencies pertain to the following sequence of frequency ratios f n / f1 = 1, 2.09, 3.17, 4.25, 5.33, 6.40, 7.48, 8.55, 9.62 and 10.70. To get a feeling of how much these values deviate from the frequencies of the harmonic series for an instrument, one may compute the corresponding musical “intervals”, in terms of the musical scales commonly used in western music. Application to the Bessel horn shown in Figure 4.16a, leads to large inharmonicity errors of intervals ranging between the quarter-tone and a full tone, for the first ten modes or so. For higher order modes, the error intervals are lower than the quarter-tone. Even so, these results are clearly unacceptable for a proper sounding instrument. Another example is shown in Figures 4.16c–d, pertaining to a Bessel horn with parameter α = 1.5 . The first ten modal frequencies of this system are about 262, 484, 706, 928, 1152, 1377, 1602, 1827, 2052 and 2278 Hz. The corresponding frequency ratios – f n / f1 = 1, 1.85, 2.69, 3.54, 4.40, 5.25, 6.11, 6.97, 7.83 and 8.69 – display a similar inadequate behaviour, with the lower-order modes deviating from the harmonic series in excess of a half-tone and up to more than one tone. Beyond the sixth mode inharmonicity errors are somewhat lower, within a half-tone. These illustrative results explain the reason why the basic bores of the wind instruments are either cylindrical or conical, while some of them, the brasses in particular, are terminated at the outlet by a fast diverging flare, used as an impedance adapter for sound radiation. It should also be mentioned that in real-life instruments, the natural frequencies are affected in an audible manner by various important geometrical details, such as the shape of the inlet mouthpiece, the tone holes and the outlet flare. As a consequence, manufacturers must refine the tuning of their instruments by minute changes in the bore profile, as well as in the position and diameter of the tone holes, or the tube length. 4.2.2
Transfer matrix method (TMM)
4.2.2.1 Transfer matrix of a uniform tube element As shown in Figure 4.17, we consider a uniform tube element of length L. Furthermore, fluid properties (mean density ρ f and speed of sound c f ) are assumed to be uniform inside the tube. For convenience, the volume velocity and pressure at the inlet of the element are denoted q1 and p1, while q2 and p2 denote the corresponding quantities at the outlet. Coming back to the principle of the modal analyses performed just above, it may be noticed that, from the mathematical standpoint, the two constants of integration A and B appearing in the general solution of the local equation [4.12] are univocally defined if both the pressure and the volume velocity are specified at the inlet of the tube element.
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Fluid-structure interaction
Figure 4.17. Uniform tube element
An elementary calculation gives: p1 = Aq1 =
iS f B ρ f cf
=
iB Z
[4.111]
Z is the acoustical impedance of the tube element. Substituting [4.111] into [4.42] and [4.43], the following matrix equation is immediately obtained: ⎡ q ( x; k ) ⎤ ⎡ q1 ⎤ ⎢ ⎥ = ⎡⎣ A ( x; k ) ⎤⎦ ⎢ ⎥ ⎣ p1 ⎦ ⎣ p ( x; k ) ⎦
[4.112]
which expresses the volume velocity q ( x; k ) and the pressure p ( x; k ) at any current point x within the element and at the wave number k = ω / c f , in terms of their values at the inlet. Accordingly, the tube element can be viewed as a “transfer box” of a linear system with two inputs and two outputs, as already explained in [AXI 04], Chapter 7. The mathematical properties of this box are specified by the transfer matrix: i ⎡ ⎤ cos kx − sin kx ⎥ ⎢ A ( x; k ) = Z ⎢ ⎥ sin cos kx ⎦ − i Z kx ⎣
[4.113]
The following important points can be easily verified: 1 Lack of symmetry of A Though the problem is conservative, the transfer matrix is not symmetrical (selfadjoint). This is a mere consequence of the choice made in the definition of the input and output vectors which mix kinematical and stress variables. Symmetric matrices can be easily obtained for instance by using the volume velocity vector [ q2 q1 ] as an input and the force vector ⎡⎣ + S f p2 − S f p1 ⎤⎦ as an output. The
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281
reason for the negative sign ascribed to the second force component is that a positive pressure at the outlet produces a positive force (traction) whereas at the inlet it produces a negative force (compression). By using [4.113], it is straightforward to show that: ⎡ cot kx ⎡ S f p2 ⎤ ⎢ ⎢ i = ⎢−S p ⎥ ⎣ f 1 ⎦ ⎢− ρ f c f ⎢⎣ sin kx
ρ f cf ⎤ sin kx ⎥⎥ ⎡ q2 ⎤ ⎢ ⎥ ⎥ q1 cot kx ⎥ ⎣ ⎦ ⎦
−
[4.114]
Nevertheless, the mixed formulation using the transfer matrices is preferred to the symmetrical formulations as they are more convenient to build numerical models of complicated pipe systems, as further explained in section 4.3.4. 2 Inverse of A As could be easily expected, A is regular (det A = 1) and its inverse is obtained simply by inverting the inlet and the outlet that is by changing the sign of x: −1
⎡⎣ A ( x; k ) ⎤⎦ = ⎡⎣ A ( − x; k ) ⎤⎦
[4.115]
3 Modal analysis using A A can be conveniently used to find the acoustical modes of the pipe, provided the suitable inlet and outlet impedances are specified. As an elementary example, let us consider the case of a tube terminated by two pressures nodes. The modal problem is written as: i ⎡ ⎤ − sin kL ⎥ ⎡ q1 ⎤ ⎡ q2 ⎤ ⎢ cos kL Z ⎢0⎥=⎢ ⎥ ⎢⎣ 0 ⎥⎦ ⎣ ⎦ −iZ sin kL cos kL ⎣ ⎦
The second row implies that sin kL = 0 , so kn L = nπ ⇔ f n =
[4.116] nc f
; n = 1, 2,... . 2L Then, the mode shapes can be expressed in terms of volume velocity, by using the first row of equation [4.116] written at a current position x instead of L. Finally, using the momentum equation [4.10], the mode shapes in terms of pressure ϕ n ( x ) = sin kn x are in full agreement with the formulas [4.75], as should be. The
method can be extended in a similar way to the case of any pair of inlet and outlet impedances to produce the same results as formulas [4.74]. This is left as a short exercise to the reader.
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Fluid-structure interaction
As further described in this and the next chapter, the transfer matrix method (in short TMM) is a powerful mathematical technique to build discretized models for performing analytical and numerical studies of the plane wave acoustics in complex pipe systems. Moreover, the TMM is not restricted to plane wave acoustics. It is also often used to analyse the linear vibrations of beam assemblies, especially in the range of short wavelengths. As shown later in this section, and then in Chapter 7, the TMM is particularly efficient to deal with plane acoustical waves in tubular circuits, because of the following aspects of the formulation: • Two scalar fields p and q are sufficient to describe the waves. • The general solution of the plane acoustical waves in a tube of constant cross-sectional area, filled with a homogeneous fluid, can be expressed in a compact and simple analytical form (equations [4.42] and [4.43]). Compact analytical solutions are also available for a few horn geometries. • In most cases of practical interest, the tubular circuits are made up of an assembly comprising a few distinct types of elements. Hence, to model the whole circuit, only a few types of transfer matrices are necessary, which are able to describe the acoustical pipe elements independently from their length, in contrast to the finite element method. The reason for the difference between the two methods is that transfer matrices are built by using exact analytical wave solutions of an element, whereas the finite element matrices are built by using low degree polynomials to approximate the actual solution within a small domain (cf. [AXI 05], Chapter 4). Therefore, the TMM models are much more economical in terms of size (number of degrees of freedom) than the FEM models. Moreover, the degree of accuracy is independent of the range of wavelengths explored, within the limits of validity of the plane wave approximation, at least. The TMM models are also advantageous when compared with the modal models, since there is no need to compute first the natural modes of vibration and then to project the local equations of motion onto the mode shapes. As demonstrated later in Chapter 7, contrasting with the finite element method, extension of the transfer matrix method to dissipative problems and terminal impedances of various kinds is straightforward and even immediate. However, because the method is formulated either in terms of frequency or Laplace variable s = iω , its range of application is strictly limited to the linear domain. 4.2.2.2 Assembling of two tube elements Let us consider a pipe made of two uniform tube elements which may differ from each other by the size and the fluid properties, see Figure 4.18. Volume velocity and pressure are defined at the ends of each element, q1 , p1 and q2 , p2 referring to the inlet and the outlet of the first element, respectively, q3 , p3 and
Plane acoustical waves in pipe systems
283
Figure 4.18. Pipe made of two distinct tube elements
q4 , p4 referring to the inlet and the outlet of the second element. In agreement with the continuity conditions [2.28] (see also [4.19]), the acoustical link between the two elements is expressed as the two holonomic conditions: q2 = q3
;
p2 = p3
[4.117]
In order to perform analytical calculations, it may be convenient to use the conditions [4.117] to eliminate the superfluous variables. In this way, one is led to describe the whole pipe by a single transfer matrix, defined as the product of the two elementary matrices [ A] = [ A2 ][ A1 ] , which yields here: ⎡ ⎡ q4 ⎤ ⎢ cos k2 L2 ⎢p ⎥ = ⎢ ⎣ 4 ⎦ −iZ sin k L ⎢⎣ 2 2 2
−
i ⎤⎡ sin k2 L2 ⎥ ⎢ cos k1 L1 Z2 ⎥⎢ cos k2 L2 ⎥⎦ ⎢⎣ −iZ1 sin k1 L1
−
i ⎤ sin k1 L1 ⎥ ⎡ q1 ⎤ Z1 ⎥⎢p ⎥ cos k1L1 ⎥⎦ ⎣ 1 ⎦
[4.118]
The coefficients of the transfer matrix [4.118] follow as: A (1,1) = cos k2 L2 cos k1 L1 − σ 12 sin k1 L1 sin k2 L2 A (1, 2 ) =
−i ( cos k2 L2 sin k1 L1 + σ 12 cos k1 L1 sin k2 L2 ) Z1
A ( 2,1) = −iZ2 ( cos k1 L1 sin k2 L2 + σ 12 cos k2 L2 sin k1 L1 ) A ( 2, 2 ) = cos k 2 L2 cos k1 L1 − σ 21 sin k1 L1 sin k2 L2 where σ 12 =
Z1 Z2
;
σ 21 =
Z2 Z1
[4.119]
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Fluid-structure interaction
4.2.2.3 Two connected tubes of distinct cross-sectional areas The acoustical modes of a pipe open at both ends must verify the equation A ( 2,1) = 0 , which yields the characteristic equation to determine the natural frequencies: 1 + σ 12 tan k1 L1 cot k2 L2 = 0
[4.120]
In the case of a pipe stopped at the inlet and open at the outlet, the matrix coefficient A ( 2, 2 ) must be zero, whence the characteristic equation: cos k1 L1 cos k2 L2 − σ 21 sin k1 L1 sin k2 L2 = 0
[4.121]
Generally, neither equation [4.120] nor [4.121] have roots ordered as a harmonic sequence. It is of interest to particularize the problem to the case of a sudden change of area and uniform fluid ( k1 = k2 = k ) . The impedances of the tube elements are Z1 = ρ f c f / S1 and Z2 = ρ f c f / S2 , respectively. The problem is further simplified by
considering two tube elements of same length L1 = L2 = L . Equation [4.120] simplifies into: −i (Z1 + Z2 ) cos kL sin kL = 0 ⇒ sin 2kL = 0
[4.122]
As in the case of the conical tube open at both ends, the natural frequencies are the same as those of a uniform tube of same length 2L and open at both ends. Concerning the pressure mode shapes, they differ or not from the uniform case depending whether there is a pressure antinode, or not, at the change of the crosssectional area of the pipe. With the aid of the transfer matrix of each tube element, the following modal pressure profiles are obtained: • first element: 0 ≤ x ≤ L ϕ n ( x ) = sin
nπ x 2L
2nπ ( x − L) n S1 ⎧ ⎪⎪ ϕ 2 n ( x ) = ( −1) S sin 2L 2 • second element: L ≤ x ≤ 2 L ⎨ (2 − 1) n π ( x − L) ⎪ϕ ( x ) = ( −1)n cos 2 n −1 ⎪⎩ 2L
[4.123]
The pressure profiles of the first two acoustical modes for S1 / S2 = 0.5 are plotted in Figure 4.19. It may be verified that the profile of the first mode has an antinode at x = L and is the same as in the uniform case. The second mode has a node at x = L where a finite discontinuity in the slope of the pressure profile is present. The jump has the necessary value to preserve the continuity of the volume velocity at the junction.
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285
Figure 4.19. Pressure profile of the two first modes for a pipe with a sudden change of crosssection and open at both ends
In the case of a pipe stopped at the inlet and open at the outlet, the characteristic equation [4.121] becomes:
( cos kL )
2
− σ 21 ( sin kL ) = 0 ⇔ tan kL = ± σ 12 2
[4.124]
The natural frequencies are ordered as the superposition of two distinct sequences corresponding to the branches + σ 21 and − σ 21 of the characteristic equation, respectively. They can be expressed analytically as: f n( ) = +
(n + θ ) c f 2L
where θ =
tan −1
; n = 0,1, 2...
(
π
σ 12
f n( ) = −
(n − θ ) c f 2L
; n = 1, 2, 3...
[4.125]
).
Figure 4.20. Pipe with a sudden change of cross-section stopped at the inlet and open at outlet: plot of the characteristic equation giving the natural frequencies
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Fluid-structure interaction
Figure 4.21. Pressure profile of the first two modes for a pipe with a sudden change of crosssection, stopped at the inlet and open at the outlet
The characteristic function, as given by the first expression in [4.124], is plotted in Figure 4.20. It refers to L = 0.5 m, σ 12 = 0.5 and air at STP ( c f 344 m/s ). The heavy dots correspond to the natural frequencies. With the numerical data specified in Figure 4.20, the fundamental of the first sequence is about 104.6 Hz, and that of the second sequence is 239.4 Hz, differing thus markedly from those of the equivalent uniform tube (fundamental 86.04 Hz and second harmonic 258 Hz). To recover a harmonic sequence of resonances, σ 21 must tend either to one, or to zero, or to infinity. Figure 4.21 displays the pressure profile of the fundamental of each sequence. Both of them have a finite discontinuity in the slope at the change of cross-sectional area. 4.2.2.4 Two connected tubes filled with distinct fluids Let us consider the circuit displayed in Figure 4.22. The impedance of the first tube element is ρ1c1 / S f and that of the second element is ρ 2 c2 / S f . Assuming the tube is open at both ends, the characteristic equation [4.120] is written as: sin
ω L2 ωL ρ c ωL ωL cos 1 + 1 1 sin 1 cos 2 = 0 c2 c1 ρ 2 c2 c1 c2
[4.126]
Depending whether the speed of sound is the same in both elements, or not, the results already established in the case of a sudden change in the cross sectional area are valid, or not. As an example let us assume that the first tube element of length L1 = 4 m is filled with a dense and poorly compressible liquid and the second tube element of equal length is filled with a light and highly compressible gas. The characteristic equation is plotted in Figure 4.23 in the frequency range 1 to 175 Hz, with the numerical values indicated in the figure.
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287
Figure 4.22. Assembly of two tube elements filled with distinct fluids
Figure 4.23. Characteristic function of the pipe filled with distinct fluids
Such results are sufficient to illustrate that the resonances of the compound system differ markedly from that of each constituent. Actually, due to the very large difference in fluid density, the natural frequencies of the water element are poorly affected by the connection to the gas element. In contrast, the resonances of the gas element are greatly modified, as could be expected a priori. It is also of interest to describe briefly the pressure profile of a few modes, as shown in Figure 4.24. The lowest frequency mode is characterized by a linear variation of pressure within the first element, as expected since at the frequency considered compressibility of the liquid column is negligible ω1 L1 / c1 0.45 . The profile of pressure in the second element is similar to that of the first quarter wavelength resonance of the tube element. This is clearly due to the large inertia of the liquid, which prevents practically any motion of the gas at the liquid-gas interface. Already at the second
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Fluid-structure interaction
mode, compressibility of the liquid becomes important and so the motion of the liquid-gas interface. Nevertheless vibration of the gas column is still largely influenced by the presence of the liquid column.
Figure 4.24. Pressure profile of the modes n =1,2,4,5
The pressure profiles of the modes n = 4 and n = 5 are suitable cases of which to discuss to what extent the modes of the liquid element is perturbed by the connection with the gas element. Actually, the first half wavelength resonance of the liquid column at 150 Hz is replaced by a pair of resonances at about 148.9 Hz and 150.8 Hz respectively. The pressure profile within the first element corresponds to that of the first half wavelength mode of the liquid column while the pressure profile
Plane acoustical waves in pipe systems
289
within the second element corresponds essentially to the second quarter wavelength mode of the gas column. As a classical result, the coupling between these two resonances gives rise to a pair of an in-phase and out-of-phase (phase opposition) modes. The pair of coupled modes is very closely spaced in frequency because of the large discrepancy between the properties of the two fluids, here the density essentially. 4.2.2.5 Helmholtz resonators
Figure 4.25. Helmholtz resonator
The remark made in the last subsection concerning the more or less compressible behaviour of a fluid vibrating at a given frequency can be further investigated by particularizing the transfer matrix [4.113] to the case of small frequencies and large wavelengths. Of course, the argument of the circular functions present in [4.113] identifies with the compressibility parameter Λa , provided the axial position x along the tube element is selected as the pertinent scale factor of length. Hence, if Λa is sufficiently small, a series expansion limited to the first order produces a simplified version of [4.113] especially useful to interpret the physical meaning of calculations performed in the range of wavelengths larger than the length of the pipes. The example of the Helmholtz resonator is particularly suitable to illustrate this important point. The device shown in Figure 4.25 comprises an enclosure of large volume (VE ) connected to a narrow tube of much smaller volume (VT ) . The enclosure is sketched as a tube element of large cross-section, (length LE , crosssectional area S E ) which is stopped at the inlet and open at the outlet. The narrow tube, of cross-sectional area ST << S E and length LT , is open at both ends. Furthermore, in the low frequency range of interest here, it is also assumed that the sound wavelengths are much larger than both LE and LT . It can be shown that, provided the fluid and the geometrical properties of the device are within an appropriate range of values which will be specified later, the first acoustical resonance of the device can be determined by using a lumped fluid model in which the fluid inside the narrow tube is modelled as an equivalent mass while the fluid
290
Fluid-structure interaction
inside the enclosure is modelled as a linear spring. Thus the Helmholtz resonance may be rightly considered as the fluid counterpart of the mass-spring system of the solid mechanics, see Figure 4.26. The Helmholtz frequency f H can be determined, as a first approximation at least, by modelling the fluid inside the narrow tube by the mass coefficient: M T = ρ f ST LT
[4.127]
Then, to determine the stiffness coefficient equivalent to the fluid in the large enclosure, it can be noticed that it must contract and expand alternately by the maximum amount δVE = ST X T , where X T denotes the amplitude of the oscillation of the fluid inside the narrow tube. As a consequence, the pressure inside the enclosure oscillates with the amplitude: p = − ρ f c 2f
ST X T S X = −E f T T VE S E LE
[4.128]
Figure 4.26. Mass-spring system equivalent to the Helmholtz resonator
The pressure force exerted on the fluid column contained in the narrow tube follows immediately as: F = −E f
ST2 X T VE
[4.129]
Therefore, the fluid within the large enclosure acts as a spring with the stiffness coefficient: KE = E f
ST2 VE
[4.130]
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291
On the other hand, inertia of the fluid contained within the enclosure can be safely neglected in comparison with that of the narrow tube provided the crosssectional area of the enclosure is much larger than that of the narrow tube. Using the partial results [4.127] and [4.130], the frequency of the Helmholtz resonance is found to be: fH =
1 2π
c c KE ST VT = f = f M T 2π VE LT 2π LT VE
[4.131]
where VT denotes the volume of the fluid within the narrow tube. As already asserted, to be valid, such a model requires, as a necessary and sufficient condition, that the wavelength of the Helmholtz resonance is much larger than LT and LE , that is: λH =
cf fH
= 2π LT
VE >> max ( LT , LE ) VT
[4.132]
Many types of physical devices may be used as a Helmholtz resonator, which means that they can be suitably described according to the lumped acoustical model presented just above, at least in the vicinity of the Helmholtz resonance. The same problem can be formulated by using the transfer matrix method, leading to more general and enlightening results than the lumped model because they include the Helmholtz resonance but are not restricted to its vicinity. 1. Transfer matrix of the narrow tube The transfer matrix of the narrow tube is formulated by expanding the matrix [4.113] to the first order with respect to ω LT / c f , which is assumed to be small. The result is the incompressible form of the transfer matrix of a tube element: 1 ⎡ ⎢ ωρ f LT AT ( LT ; ω ) = ⎢ −i ST ⎣⎢
0⎤ ⎥ 1⎥ ⎦⎥
[4.133]
In agreement with the results already established in Chapter 2, subsection 2.2.2, the volume velocity is unchanged and the pressure varies linearly along the tube. As a consequence, it can be stated that the coefficient AT ( 2,1) of the transfer matrix accounts for the fluid inertia. 2. Transfer matrix of the large enclosure To the first order in the compressibility parameter, the matrix [4.113] simplifies into the compressible form:
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Fluid-structure interaction
⎡ 1 ⎢ AE ( LE ; ω ) = ⎢ ⎢ iωρ f LE ⎢− SE ⎣⎢
−
iωVE ⎤ ρ f c 2f ⎥ ⎥ ⎥ 1 ⎥ ⎦⎥
[4.134]
The coefficient AE ( 2,1) accounts for the fluid inertia within the enclosure, which was neglected in the simplistic lumped model. Of course, this term can be safely neglected as soon as VE is sufficiently large, or, to be more specific, if the following inequality holds: AE ( 2,1) << AT ( 2,1) ⇔
LT L >> E ST SE
[4.135]
On the other hand, because of the non vanishing coefficient AE (1, 2 ) , proportional to VE , the volume velocity changes from the tube inlet to the tube outlet. This of course is a necessary feature of the model, since the tube element is stopped at the inlet and the fluid must oscillate at the connection with the second element. Then, the coefficient AE (1, 2 ) accounts for the fluid compressibility within the large enclosure. X T denoting once more the fluid displacement at the interface between the two tube elements, the volume velocity is there qT = iω ST X T . Identifying this value with that arising from the transfer matrix [4.134], the result [4.129] is recovered and so the final formula [4.131]. 3. Transfer matrix of the Helmholtz resonator The matrix product of [4.133] and [4.134] provides the transfer matrix characterizing the Helmholtz resonator as a whole, which is conveniently written as: ⎡ ⎢ [ AH ] = [ AT ][ AE ] = ⎢⎢ ⎢ −iωρ f ⎣⎢
iωVE ⎤ ⎡ ρ f c 2f ⎥ ⎢ ⎥=⎢ 2 ⎥ ⎢ ⎛ LT LE ⎞ ω ⎜ + ⎟ 1 − 2 ⎥ ⎢ −ik ρ f c f ω H ⎦⎥ ⎣⎢ ⎝ ST S E ⎠ 1
−
ikVE ⎤ ρ f cf ⎥ ⎥ 2⎥ ⎛ LT LE ⎞ ⎛ k ⎞ ⎜ + ⎟ 1− ⎜ ⎟ ⎥ S S E ⎠ ⎝ T ⎝ k H ⎠ ⎦⎥ 1
−
[4.136] where again ω H is the natural pulsation of the Helmholtz resonance. As described later, Helmholtz resonators are often used in industry for the acoustic isolation of various piping systems. In that context, it is important to be able to build resonators of reasonable size and low Helmholtz frequency. A particularly interesting configuration combines a tubular circuit filled with a liquid connected to an enclosure filled with a gas. Using the transfer matrices [4.133] and [4.134], it may be easily shown that the Helmholtz frequency of the system is:
Plane acoustical waves in pipe systems
fH =
1 2π LL
EGVL ρ LVG
293
[4.137]
where LL is the length of that part of the circuit which is filled with liquid (density ρ L ) and EG is the Young modulus of the gas. Finally VL and VG are the volumes occupied by the liquid and the gas, respectively. As an example, let us calculate the Helmholtz frequency corresponding to the following numerical data:
water: LL = 30 m; S L = 0.0π m 2 ; cL = 1200 ms-1; ρ L = 1000 kgm-3 air: LG = 2 m; SG = π m 2 ; cG = 344 ms-1; ρG = 1.2 kgm-3 If the whole circuit is filled with water, the Helmholtz frequency is found to be about 2.466 Hz and the wavelength is about 490 m. If the enclosure is filled with air at STP, the Helmholtz frequency is decreased by two orders of magnitudes and the wavelength is correlatively increased by the same factor. 4.2.2.6 Higher plane wave modes of an enclosure tube assembly So long as LT is larger than LE , when the frequency range is sufficiently increased beyond the Helmholtz resonance, compressibility of the fluid inside the narrow tube becomes important, while the enclosure can still be suitably described by using the lumped model [4.134]. In that domain, the circuit is governed by the following transfer matrix equation: ⎡q ⎤ ⎢ out ⎥ ⎢⎣ 0 ⎥⎦
=
⎡ cosω LT ⎢ cT ⎢ ⎢ −i ρ c T T ⎢ sin ω LT cT ⎢⎣ ST
⎤ ⎤⎡ 1 − iωVE2 ⎥ ⎡ ⎤ − iST sin ω LT ⎥ ⎢ cT ⎥ ⎢ ρT cT ⎥ ρ E cE ⎢ 0 ⎥ ⎥ p ⎥⎢ L ω ⎥ ⎢⎣ in ⎥⎦ cos T ⎥ ⎢ −iωρ E LE 1 ⎥ cT ⎥⎦ ⎢ S E ⎣ ⎦
[4.138]
With the aid of a few elementary manipulations the characteristic equation giving the natural frequencies of the tube enclosure assembly is suitably written as:
A22 = 0 ⇒ cos
ω LT ⎛ ω LE ⎞ ⎛ ZT ⎞ ω LT −⎜ =0. ⎟ ⎜ ⎟ sin cT cT ⎝ cE ⎠ ⎝ ZE ⎠
[4.139]
where ZT and ZE are the impedances of the tube and enclosure, respectively. If the problem is further particularized to the case of a uniform fluid, [4.139] can be written as:
294
Fluid-structure interaction
ϖ tgϖ =
VT VE
where ϖ =
[4.140] ω LT . c0
The pressure mode shapes within the tube are found to be: ⎛ϖ x ⎞ ⎛V ϕ n ( x ) = cos ⎜ n ⎟ − ϖ n ⎜ E L ⎝ T ⎠ ⎝ VT
⎞ ⎛ϖnx ⎞ ⎟ sin ⎜ ⎟ ⎠ ⎝ LT ⎠
[4.141]
Figure 4.27. Characteristic function of the enclosure-tube assembly
Equation [4.140] is represented graphically for a volume ratio VT /VE = 0.1 in Figure 4.27. The first natural frequency corresponds to the Helmholtz frequency [4.137]. The pressure profile of the Helmholtz resonance is shown in Figure 4.28. The pressure is uniform within the enclosure and decreases linearly within the tube. The infinite many other roots of equation [4.140] tend very rapidly towards the values nπ , n = 1, 2,... , and to the mode shapes ϕ n ( x ) = sin (ϖ n x / LT ) , that is the acoustic modes of the narrow tube open at both ends. Hence it is verified that the connection of a tube to a large volume of the same fluid is practically equivalent to a pressure node for all the modes, except the Helmholtz mode.
Plane acoustical waves in pipe systems
295
Figure 4.28. Pressure profile of the first mode of the tube-enclosure assembly
4.2.2.7 Enclosure-tube assembly: case of a very short tube
Figure 4.29. Large uniform tube connected to a small uniform tube
It may be of interest to consider the special case where the narrow tube is also much shorter than the tube element simulating the enclosure, see Figure 4.29. This kind of system has been already considered in Chapter 2 subsection 2.2.2, for an incompressible fluid. It was concluded that the inertia of the fluid contained within the small tube can be accounted for by an inertial impedance and reciprocally, the impedance associated with an orifice can be modelled as a small tube. As could be expected, the acoustical modes of the large tube modelling the enclosure can be significantly modified by such inertial effects. This justifies in particular the care taken by the manufacturer of wind instruments in the appropriate size, and even shape, of the lateral bores. Using the notations of Figure 4.29, the general equation [4.119] particularized to the case ω / c f << 1 gives the following characteristic equation:
296
Fluid-structure interaction
A ( 2, 2 ) = cosϖ − κϖ sin ϖ = 0
where κ =
Sf sf L
and ϖ =
[4.142]
ωL . cf
Depending whether the geometrical factor κ is small or large, the natural frequencies tend to those of the quarter wavelengths modes (stopped-open tube), or to the half wavelengths modes (tube stopped at both ends). On the other hand, for a given geometry, as the modal rank increases, the corresponding natural frequency tends to that of the tube stopped at both ends. Such results are not surprising since the fluid contained in the small tube behaves as an added mass proportional to κ and the inertial effect increases as the frequency squared. Of course, the same results can be recovered by modelling the small tube as a terminal impedance given by the formula [4.60]. The latter can be easily derived starting from the approximate form [4.133]: ⎡ 1 ⎡ q( L + ) ⎤ ⎢ ⎢ 0 ⎥ = ⎢ −i ωρ f ⎣ ⎦ ⎢ sf ⎣
0⎤ ⎡ 1 ⎡ q( L ) ⎤ ⎢ ⎥ ⎡ q( L ) ⎤ ωρ f ⇔⎢ = 1⎥ ⎢⎣ p ( L) ⎥⎦ p( L) ⎥⎦ ⎢ +i ⎣ ⎥⎦ ⎢⎣ sf
0⎤ ⎥ ⎡ q( L + ) ⎤ 1⎥ ⎢⎣ 0 ⎥⎦ ⎥⎦
[4.143]
whence:
Zout =
iρ f c f Sf
tan αout =
p( L) iωρ f ω S f = ⇒ tan αout = q( L ) sf cf sf
To conclude this subsection it is worthwhile noticing that the principle of the whole family of Helmholtz resonators where the narrow tube is replaced simply by an orifice is precisely based on the inertia of the fluid oscillating in the vicinity of the hole. 4.3. Forced waves 4.3.1
Concentrated acoustical sources
4.3.1.1 Volume velocity (monopole) source The volume velocity source was introduced in Chapter 1, subsection 1.2.2.2 as an external source of fluid material, accounted for as a term in the right-hand side of the mass equation. Here, it is assumed that the source is concentrated at a given cross-section. Hereafter, the theory is presented in terms of curvilinear coordinate denoted s, to stress that it holds for curved as well as for straight tubes. The length 2ε of the tube shown in Figure 4.30 is assumed to be infinitesimal, in such a
Plane acoustical waves in pipe systems
297
Figure 4.30. Volume velocity source located at the junction of two tube elements of infinitesimal length
manner that it can be considered as uniform, at least on that local scale. The positions of the inlet and outlet are identified by using the curvilinear abscissas s1 and s2 respectively. Since ε is infinitesimal, the mass flow rate and the volume velocity are the same at the outlet as at the inlet:
b g b g
q s2 − q s1 = 0
[4.144]
Starting from this configuration, let us cut mentally the tube by a cross-section located at midspan, identified by the curvilinear abscissa: s0 = s1 + ε = s2 − ε
[4.145]
Let us assume that some fluid is injected into or removed out of the pipe at so , according to the fluctuating mass flow rate: dM f dt Q(
e)
= ρ f Q(
(t )
e)
(t )
[4.146]
is the volume velocity of the external source (or sink) of fluid. The mass
equation applied to the control volume defined by the tube element reads as: q ( s2 ) − q ( s1 ) = Q (
e)
(t )
[4.147]
The external source of fluid can be expressed as a Dirac distribution put on the right-hand side of the mass equation. The momentum equation is not modified. Thus the forced waves are found to be governed by the following equations, written here in terms of time instead of pulsation: Sf ∂ p ∂q e +ρf = ρ f Q ( ) ( t ) δ ( s − so ) 2 ∂s cf ∂ t
[4.148]
298
ρf
Fluid-structure interaction
∂q ∂p + Sf =0 ∂t ∂s
[4.149]
Eliminating q between the two equations [4.148] and [4.149] is immediate, leading to the wave equation forced by an external volume velocity source concentrated at s0 : e Sf ∂ 2 p ∂ Q( ) (t ) ∂ ⎛ Sf ∂ p ⎞ = − δ ( s − s0 ) ⎜⎜ ⎟⎟ − ∂ s ⎝ ρ f ∂ s ⎠ ρ f cf 2 ∂ t 2 ∂t
[4.150]
It is recalled that equation [4.150] written in terms of distributions is equivalent to the forced system written in terms of ordinary functions (cf. [AXI 05] Chapter 3): Sf ∂ 2 p ∂ ⎛ Sf ∂ p ⎞ =0 ⎜⎜ ⎟⎟ − ∂ s ⎝ ρ f ∂ s ⎠ ρ f cf 2 ∂ t 2 Sf ⎛∂ p ⎜ ρf ⎝ ∂ s
∂p − s = s0+ ∂s
e ⎞ ∂ Q( ) (t ) = − ⎟ s = s0− ∂t ⎠
[4.151]
Therefore, the concentrated volume velocity source imposes a finite discontinuity of the volume velocity located at the position of the source. The fact that the acoustical source is proportional not to the injected volume velocity but to the time derivative of it means that no sound can be excited by a steady flow, in agreement with the common observation that a torrent is far more noisy than a peaceful river. The physical reason is a mere consequence of the inertia principle of Galileo, according to which no force, hence no pressure, can be induced by a rectilinear and uniform motion. In particular, in order to produce a sound in a wind instrument, the essentially constant flow rate exhaled by the lungs of the player must be converted into an oscillating flow rate. For that purpose the air stream is suitably chopped either by using a mechanical reed like a piece of flexible cane or by letting the air stream flips alternatively in and out of a hole. Actually, the physical mechanism governing such a conversion of the air flow to produce a musical sound involves subtleties which are beyond the scope of the present book. Either in the case of the mechanical reed or the flipping air stream, the dynamical behaviour of the instrument involves a nonlinear coupling between the source and the response of the instrument. For instance, in the case of a flute, the player blows an air stream impinging more or less tangentially the mouth hole. The coupling between the “permanent” flow produced by the player and the acoustical resonances of the tube is so efficient that the flow is strongly modified and oscillates with a large amplitude at a frequency close to the tube resonance. The mechanism, broadly known as acoustic locking, or “accrochage”, is one type of coupling which can occur between an oscillation and a permanent flow. Such mechanisms will be the object of the last book in the present series.
Plane acoustical waves in pipe systems
299
Finally, it is useful to mention that in the literature devoted to acoustics the concentrated volume velocity source is traditionally termed a monopole source. The reason will be made clear in the next subsection where it will be found that a concentrated pressure source, traditionally termed a dipole source, can be described in terms of a Dirac dipole. Extension to the two and three dimensional cases and to multipole sources is postponed to Chapter 5. 4.3.1.2 Pressure (dipole) source If a fluctuating finite pressure jump P (
e)
(t )
is imposed at so , the momentum
equation becomes: ρf
∂q ∂p e + Sf = S f P ( ) ( t ) δ ( s − so ) ∂t ∂s
[4.152]
Eliminating q between the mass equation and the forced momentum equation [4.152] yields the forced wave equation written as: ( ) ( ) ∂ ⎛ S f ∂ p ⎞ S f ∂ 2 p S f P ( t ) d δ ( s − so ) S f P ( t ) = = δ ′ ( s − so ) ⎜⎜ ⎟⎟ − 2 2 ∂ s ⎝ ρ f ∂ s ⎠ ρ f cf ∂ t ρf ρf ds e
b
g
e
[4.153]
where δ ′ s − so is the Dirac dipole (cf. [AXI 05], Chapter 3). It is recalled that the
b
g
action of δ ′ s − so on an ordinary function P ( s
⌠ 2 ⎮ ⎮ ⎮ ⎮ ⌡s1
⎧ dP ( e ) ⎪− ∂δ − s s ( ) e o P ( ) ( s; t ) ds = ⎨ ds ∂s ⎪ 0 ⎩
e)
( s; t )
if s0 ∈ [ s1 , s2 ] s0
is:
[4.154]
otherwise
The equation [4.153] in terms of distributions is equivalent to the forced system written in terms of ordinary functions: ∂ ⎛ Sf ∂ p ⎞ Sf ∂ 2 p =0 ⎜ ⎟− ∂ s ⎜⎝ ρ f ∂ s ⎟⎠ ρ f c 2f ∂ t 2 p (s
. o+
4.3.2
[4.155]
) − p (s ) = P (s ;t) o−
e
o
Transfer functions for a uniform tube
Let us consider the uniform tube shown in Figure 4.31, excited by a harmonic e Q and/or P ( ) source concentrated at x0 . The acoustical equations [4.150] and [4.153] may be particularized as: (e)
300
Fluid-structure interaction
iωρ f ( e) ∂2 p ω2 e + 2 p = P ( )δ ′ ( x − xo ) − Q δ ( x − xo ) 2 ∂x cf Sf
[4.156]
As shown below, the response of the tube can be expressed either in an analytical compact form by using the transfer matrix method, or in terms of a series by using the modal expansion method. It is of interest to present both of these procedures to point out several aspects of theoretical and practical interest.
Figure 4.31. Uniform tube excited by acoustical sources concentrated at x0
4.3.2.1 Transfer matrix method The transfer matrix method can be conveniently used to discretize the equation [4.156] in accordance with the circuit diagram of Figure 4.32. The pipe is discretized by two uniform tube elements. The transfer matrix of the element between the inlet and the source is denoted [ Ain ] and that of the element between the source and the outlet is denoted [ Aout ] . The acoustic vector at the inlet of the
second element is related to that at the outlet of the first element by the following finite jump condition: e ⎡ q ( x0 + ; k ) ⎤ ⎡ q ( x0 − ; k ) ⎤ ⎡Q ( ) ( k ) ⎤ ⎥ ⎢ ⎥=⎢ ⎥ + ⎢ (e) ⎣ p ( x0 + ; k ) ⎦ ⎣ p ( x0 − ; k ) ⎦ ⎢⎣ P ( k ) ⎥⎦
[4.157]
The relation [4.157] serves to connect the two elements. The acoustic vectors from one side to the other of the source can be expressed in terms of their values at the inlet and outlet respectively: ⎡ q ( x0− ; k ) ⎤ ⎡ q ( 0; k ) ⎤ ⎡1⎤ ⎢ ⎥ = ⎡⎣ Ain ( kx0 ) ⎤⎦ ⎢ ⎥ = qin ⎡⎣ Ain ( kx0 ) ⎤⎦ ⎢ ⎥ ⎣Zin ⎦ ⎣ p ( x0− ; k ) ⎦ ⎣ p ( 0; k ) ⎦ ⎡ q ( x0+ ; k ) ⎤ −1 ⎡ q ( L; k ) ⎤ −1 ⎡ 1 ⎤ ⎢ ⎥ = ⎣⎡ Aout ( kL )⎦⎤ ⎢ ⎥ = qout ⎣⎡ Aout ( kL ) ⎦⎤ ⎢ ⎥ ⎣Zout ⎦ ⎣ p ( x0+ ; k ) ⎦ ⎣ p ( L; k ) ⎦
[4.158]
Plane acoustical waves in pipe systems
301
Figure 4.32. Transfer matrix model
Whence the forced matrix equation: (e) ⎡ 1 ⎤ ⎡ 1 ⎤ ⎡Q ⎤ qout ⎣⎡ Aout ( kL ) ⎤⎦ ⎢ ⎥ − qin ⎣⎡ Ain ( kx0 ) ⎦⎤ ⎢ ⎥ = ⎢ e ⎥ ⎣Zout ⎦ ⎣Zin ⎦ ⎢⎣ P ( ) ⎥⎦ −1
[4.159]
It may be noticed that equation [4.159], which is solved to determine the independent variables qin and qout , holds even if the properties of the inlet and the outlet elements are distinct. However, in the present case the matrices of the elements are written as: sin kx ⎤ ⎡ + cos kx ⎡⎣ Ain ( kx )⎤⎦ = ⎢ iZ ⎥ ⎢ ⎥ ⎣ −iZ sin kx cos kx ⎦
0 ≤ x ≤ x0
sin k ( x − x0 ) ⎤ ⎡ cos k ( x − x0 ) + ⎥ ⎡⎣ Aout ( kx )⎤⎦ = ⎢ iZ ⎢ ⎥ ⎢⎣ −iZ sin k ( x − x0 ) cos k ( x − x0 ) ⎥⎦ sin k ( x0 − x ) ⎤ ⎡ −1 cos k ( x0 − x ) + ⎢ ⎥ ⎡⎣ Aout ( kx )⎤⎦ = iZ ⎢ ⎥ ⎣⎢ −iZ sin k ( x0 − x ) cos k ( x0 − x ) ⎦⎥
x0 ≤ x ≤ L
[4.160]
x0 ≤ x ≤ L
Using the first equation [4.158] written at a current position 0 ≤ x ≤ x0 , and the impedance relations [4.45] and [4.46], the response in the inlet tube element is expressed as: ⎡ cos ( kx + αin ) ⎤ sin kx ⎤ ⎢ ⎥ ⎡ cos αin 1 ⎡ q ( x; k ) ⎤ + ⎥ ⎢ cos kx ⎥⎡ ⎤ = q ⎢ = q Z i ⎢ ⎥ in in ⎢Z ⎥ ⎢ ⎥ ⎢ ⎥ p x ; k ( ) + α sin kx ( ⎣ in ⎦ in ) ⎣ ⎦ ⎣ −iZ sin kx cos kx ⎦ ⎢ −iZ ⎥ cos αin ⎥⎦ ⎢⎣
[4.161]
302
Fluid-structure interaction
In a similar manner, using the second equation [4.158], written at a current position x0 ≤ x ≤ L , the response in the outlet tube element is expressed as: sin k ( x − L ) ⎤ ⎡ ⎡ q ( x; k ) ⎤ + ⎢ cos k ( x − L ) ⎥⎡ 1 ⎤ ⇒ q = Z i ⎢ ⎥ out ⎢ ⎥ ⎢⎣Zout ⎥⎦ ⎣ p ( x; k ) ⎦ ⎢⎣ −iZ sin k ( x − L ) cos k ( x − L ) ⎥⎦ ⎡ cos ( k ( x − L ) − α out ) ⎤ ⎢ ⎥ cos αout ⎡ q ( x; k ) ⎤ ⎢ ⎥ q = ⎢ ⎥ out ⎢ p x k ; ( ) sin ( k ( x − L ) − α out ) ⎥ ⎣ ⎦ ⎢ −iZ ⎥ ⎢⎣ ⎥⎦ cos α out
[4.162]
Let us consider first the case of a pressure source. Substituting the matrices [4.160] into the equation [4.159], the linear algebraic system follows: qout
cos ( k ( x0 − L ) − αout ) cos αout
− qout
− qin
sin ( k ( x0 − L ) − α out ) cos αout
cos ( kx0 + α in ) =0 cos αin
sin ( kx0 + αin ) P ( e ) + qin = cos αin iZ
[4.163]
Whence: qout =
e P ( ) cos ( kx0 + αin ) cos α out iZ sin ( kL + α in + α out )
e P ( ) cos ( k ( x0 − L ) − αout ) cos α in qin = iZ sin ( kL + αin + αout )
[4.164]
Substituting [4.164] into the relations [4.161] and [4.162], the response of the pipe to a concentrated pressure source of wave number k is found to be: – in the inlet tube element 0 ≤ x ≤ x0 : ⎛ P ( e ) ⎞ cos ( k ( L − x0 ) + α out ) cos ( kx + αin ) q ( x; k ) = ⎜ ⎜ iZ ⎟⎟ sin ( kL + α in + αout ) ⎝ ⎠ e cos ( k ( L − x0 ) + α out ) sin ( kx + α in ) p ( x; k ) = − P ( ) sin ( kL + α in + αout )
– in the outlet tube element x0 ≤ x ≤ L :
[4.165]
Plane acoustical waves in pipe systems
⎛ P ( e ) ⎞ cos ( k ( L − x ) + αout ) cos ( kx0 + α in ) q ( x; k ) = ⎜ ⎜ iZ ⎟⎟ sin ( kL + α in + α out ) ⎝ ⎠ e sin ( k ( L − x ) + α out ) cos ( kx0 + α in ) p ( x; k ) = + P ( ) sin ( kL + α in + αout )
303
[4.166]
Comparing [4.165] and [4.166], it can be immediately checked that volume velocity is continuous when crossing the source position q ( x0− ; k ) = q ( x0+ ; k ) while pressure jumps by the amount p ( x0 + ; k ) − p ( x0− ; k ) = P ( ) , as appropriate. The e
corresponding transfer functions are immediately obtained by assuming P ( ) = 1 . They are written as: e
• In the inlet tube element 0 ≤ x ≤ x0 : ⎛ 2π f ( L − x0 ) ⎞ ⎛ 2π fx ⎞ + αout ⎟ cos ⎜ + α in ⎟ cos ⎜ ⎜ ⎟ ⎜ ⎟ cf P ⎝ ⎠ ⎝ cf ⎠ H q( ) ( x, x0 ; f ) = ⎛ 2π fL ⎞ + αin + α out ⎟ iZ sin ⎜ ⎜ c ⎟ ⎝ f ⎠
[4.167]
⎛ 2π f ( L − x0 ) ⎞ ⎛ 2π fx ⎞ + αout ⎟ sin ⎜ + α in ⎟ cos ⎜ ⎜ ⎟ ⎜ ⎟ cf P ⎝ ⎠ ⎝ cf ⎠ H p( ) ( x, x0 ; f ) = − sin ( kL + αin + α out )
• In the outlet tube element x0 ≤ x ≤ L : ⎛ 2π f ( L − x ) ⎞ ⎛ 2π fx0 ⎞ + αout ⎟ cos ⎜ + αin ⎟ cos ⎜ ⎜ ⎟ ⎜ c ⎟ cf P ⎝ ⎠ ⎝ f ⎠ H q( ) ( x, x0 ; f ) = ⎛ 2π fL ⎞ + αin + αout ⎟ iZ sin ⎜ ⎜ c ⎟ ⎝ f ⎠ ⎛ 2π f ( L − x ) ⎞ ⎛ 2π fx0 ⎞ + αout ⎟ cos ⎜ + αin ⎟ sin ⎜ ⎜ ⎟ ⎜ ⎟ cf P ⎝ ⎠ ⎝ cf ⎠ H p( ) ( x, x0 ; f ) = + ⎛ 2π fL ⎞ + αin + αout ⎟ sin ⎜ ⎜ c ⎟ ⎝ f ⎠
[4.168]
It is of interest to illustrate graphically the behaviour of such transfer functions. As an example, we consider a circular cylindrical tube length L = 1 m, bore radius R= 7 mm, filled with air at STP. The source is assumed to be located at x0 = 0.5L and the response is calculated at x1 = 0.4 L . Finally, the tube is assumed to be terminated at both ends by a pressure node. Anticipating the presentation made later in Chapter 7,
304
Fluid-structure interaction
(P)
( x0 , x1 ; f )
(P)
( x0 , x1 ; f )
Figure 4.33. Squared modulus of the transfer function H p
Figure 4.34. Squared modulus of the transfer function H q
the terminal impedances are provided with a small imaginary part αin = αout = 0.04i to avoid singularities at the resonance frequencies. The quantities H p(
P)
2
and H q(
P)
2
are plotted in Figures 4.33 and 4.34 in the
frequency range 0 – 4 kHz. It may be verified that only the even resonances are
Plane acoustical waves in pipe systems
305
excited, that is those modes which have a pressure node and a volume velocity antinode at the pressure source location. Furthermore, the magnitude of the resonance peaks differs greatly from one mode to the other and the largest peaks in pressure correspond to the smallest in volume velocity. The acoustical field excited by a monochromatic pressure source of frequency P P f 0 is illustrated in Figure 4.35 where the real part of H p( ) ( x; f 0 ) and H q( ) ( x; f 0 ) are plotted versus x. Here, dissipation is neglected ( αin = αout = nπ ). In the quasistatic range, fluid compressibility is practically negligible. So, the pressure field varies linearly within each tube element jumping from –0.5 to +0.5 when crossing the source. The volume velocity is constant to the first order of k0 L . However, as indicated in the plot presented here, according to the “exact calculation”, a slight variation may be brought in evidence. As could be expected, q ( x; f 0 ) is a continuous function; however, its first derivative has a finite discontinuity at the source location, in agreement with the mass equation.
(a) quasi-static excitation: f 0 << f1 = 173Hz
(b) nearly resonant excitation f 0 f 2 = 346 Hz Figure 4.35. Pressure and volume velocity fields excited by a harmonic pressure source
306
Fluid-structure interaction
(P)
Figure 4.36. Squared modulus of the transfer functions H p
( 0, 0; f )
(P)
and H q
( 0, 0; f )
Finally, Figure 4.36 displays the squared modulus of the transfer functions at the inlet for an impulsive source located at the tube inlet. All the modes are excited with the same efficiency. Comparing Figures 4.34 and 4.36, it can be concluded that if the pressure source is located at the tube inlet, the response is periodic at the frequency f1 = 173Hz , whereas if the pressure source is located at the middle of the tube, the response is periodic at the frequency f 2 = 346 Hz . As a consequence, in most wind instruments like the recorder, or the trumpet, it is advantageous to locate the mouth at the tube inlet and not at an intermediate place along the tube.
Plane acoustical waves in pipe systems
(Q )
Figure 4.37. Squared modulus of H p
( 0.5, 0.4; f ) and
H q(
Q)
307
( 0.5, 0.4; f )
The case of a source of volume velocity can be treated in the same way as that of a pressure source and there is no need to detail the mathematical manipulations which lead to the following results: • in the inlet tube element 0 ≤ x ≤ x0 :
308
Fluid-structure interaction
q ( x; k ) = −Q (
e)
sin ( k ( L − x0 ) + αout ) cos ( kx + αin )
p ( x; k ) = +iZ Q
sin ( kL + αin + αout )
(e)
sin ( k ( L − x0 ) + αout ) sin ( kx + αin )
[4.169]
sin ( kL + αin + αout )
• in the outlet tube element x0 ≤ x ≤ L : q ( x; k ) = Q (
e)
cos ( k ( L − x ) + αout ) sin ( kx0 + αin )
p ( x; k ) = +iZ Q
sin ( kL + αin + αout )
(e)
sin ( k ( L − x ) + αout ) sin ( kx0 + αin )
[4.170]
sin ( kL + αin + αout )
Figure 4.38. Pressure and volume velocity fields excited by a harmonic volume velocity source
These results are illustrated graphically considering the same pipe as in the case of the pressure source. H p(
Q)
2
and H q(
Q)
2
are plotted in Figure 4.37. In contrast to
the case of a pressure source, only the odd resonances are excited, that is those modes which have a pressure antinode and a volume velocity node at the source
Plane acoustical waves in pipe systems
309
location. Again, the magnitude of the resonance peaks highly differs from one mode to the other and the largest peaks in pressure correspond to the smallest in volume velocity. The acoustical field excited by a monochromatic source of frequency f 0 is illustrated in Figure 4.38. As expected, the pressure field is continuous and its slope, like the volume velocity, has a finite jump at the source crossing. In the quasi-static range, the pressure varies linearly in each tube element and the volume velocity is essentially uniform. Near a resonance, due to the large amplification factor of the response, the discontinuities at the source are barely detectable in the scale of the figures. 4.3.2.2 Modal expansion method The response at x1 to an acoustic source located at position x0 can be expanded in a modal series, as in the case of any linear and conservative mechanical system. The series can be expressed in terms of either time, or pulsation ω and even the Laplace variable. Choosing ω as the variable, the Fourier transform of the pressure field is expanded as: p p ( x; ω ) = ∑ an (ω ) ϕ n( ) ( x ) [4.171] n
Once more the subscripted hat symbol is used to denote the Fourier transform. The pressure mode shapes verify the norm and orthogonality relations: L
⌠ ⎮ ⎮ ⎮ ⌡0
(ϕ ( ) ( x )) p
n
2
L
dx = Lμn
;
⌠ ⎮ ⎮ ⌡0
ϕ n(
p)
( x ) ϕ m( p ) ( x ) dx = 0
if m ≠ n
[4.172]
where the coefficient μ n depends on the norm and the shape of the modes. Substituting the expansion [4.171] into the forced equation [4.150] and performing the projection onto the modal basis the following response is obtained: e e iωρ f c 2f Q p ( x0 , x1 ; ω ) = Q ( ) (ω ) H p( ) ( x0 , x1 ; ω ) = Q ( ) (ω ) Sf L e e ⎛ −c 2 Q q ( x0 , x1 ; ω ) = Q ( ) (ω ) H q( ) ( x0 , x1 ; ω ) = Q ( ) (ω ) ⎜ f ⎜ L ⎝
∞
∑ n =1
ϕ n( p ) ( xo ) ϕ n( p ) ( x1 ) μn (ωn2 − ω 2 )
⎞ ∞ ϕ n( p ) ( xo ) ϕ n′( p ) ( x1 ) ⎟⎟ ∑ 2 2 ⎠ n =1 μn (ωn − ω )
[4.173]
The volume velocity response is deduced from the pressure response by using the momentum equation, which implies a term by term derivation of the pressure series with respect to x at the response location x1 . In the same manner, substituting the expansion [4.171] into the forced equation [4.153] we obtain:
310
Fluid-structure interaction
e e c2 P p ( xo , x1 ; ω ) = P ( ) (ω ) H p( ) ( xo , x1 ; ω ) = P ( ) (ω ) f L
∞
∑
ϕ n′(
n =1
⎛ −ic 2f S f e e P q ( xo , x1 ; ω ) = P ( ) (ω ) H q( ) ( xo , x1 ; ω ) = P ( ) (ω ) ⎜ ⎜ ωρ L f ⎝
p)
( xo ) ϕ n( p ) ( x1 )
μn (ωn2 − ω 2 )
⎞ ∞ ϕ n′( p ) ( xo ) ϕ n′( p ) ( x1 ) ⎟⎟ ∑ 2 2 ⎠ n =1 μn (ωn − ω )
[4.174] Depending on the particularities of the problem treated, the series can be truncated in accordance with the criteria presented in [AXI 05], Chapter 4. On the other hand, the series [4.173] and [4.174] are very convenient to use to point out the following properties of the pressure and volume velocity responses: • Dissipation can be accounted for by using modal damping ratios ς n to transform the modal denominators into the damped form: Dn (ω n , ω ) = μ n (ω n2 − ω 2 + 2iς nω nω )
[4.175]
• The response to a volume velocity source vanishes if it is located at a p pressure node ϕ n( ) ( xo ) = 0 , and the response to a pressure source vanishes if it is located at a pressure antinode ϕ n′(
p)
( xo ) = 0 , in full agreement with the
results presented in the preceding subsection. Accordingly, to excite all the resonances of a pipe, either an impulsive pressure source located at a pressure node, or an impulsive source of volume velocity located at a stopped end, must be used. In this respect, it may be worthwhile mentioning that the flipping air stream used as a source in a flute is located at a hole, hence a pressure node, while the mechanical read used in a clarinet is located at a stopped end, hence a pressure antinode. • The contribution of a given mode to the pressure response is maximum at the place where the volume velocity response vanishes. • Inverse Fourier transformation of the results [4.173] and [4.174] allows one to shift from the spectral to the time domain and, in particular, to obtain the Green functions, i.e. the response to impulsive and concentrated acoustical sources of unit magnitude. More generally, as already emphasized in [AXI 05] in the context of structural dynamics, modal expansion methods in the time domain can be used to deal with many nonlinear problems, provided the modal density is not too high, as further discussed in Chapter 5. 4.3.3
Acoustical isolation of a piping system
By inserting a large enclosure, or cavity, in a pipe system, the impedances can be highly modified and so the acoustical modes. Actually, a large enclosure is the fluid
Plane acoustical waves in pipe systems
311
counterpart of the release of a rigid support in solid mechanics. Therefore, large enclosures can be used to isolate a circuit from undesirable noise sources such as those induced by a rotating pump or the turbulence of a permanent flow, as will be described in Volume 4 of this series. It is of interest to understand the principle of this method of acoustic isolation, which presents marked analogies with the flexible supports used to isolate a structure from external vibration sources. In both cases, the basic idea is to introduce into the system a low frequency mode of vibration, which works as a low pass filter. Here the Helmholtz resonance stands for such a filtering mode. 4.3.3.1 Cavity inserted in series with the main circuit Figure 4.39 shows a circuit comprising a single tube element. A pressure source is located at the inlet. The response could be obtained directly by using the formulas [4.166]. However, having in mind to calculate the response of the modified pipe, it is preferred here to reformulate the problem without specifying the coefficients of the transfer matrix. The acoustic vector at the tube inlet is written as: ⎡ q ( 0 ) ⎤ ⎡1 0⎤ ⎡ qin ⎤ ⎡ 0 ⎤ ⎢ ⎥=⎢ ⎥ + ⎢ (e) ⎥ ⎥⎢ ⎣ p ( 0 ) ⎦ ⎣0 1⎦ ⎣ pout ⎦ ⎣ P ⎦
[4.176]
Figure 4.39. Acoustic circuit
The response at the outlet is governed by the following equation: ⎡ q ( L ) ⎤ ⎡ A (1,1) A (1, 2 ) ⎤ ⎡ q ( 0 ) ⎤ ⎢ ⎥=⎢ ⎥⎢ ⎥⇔ ⎣ p ( L ) ⎦ ⎣ A ( 2,1) A ( 2, 2 ) ⎦ ⎣ p ( 0 ) ⎦ ⎡ A (1,1) + Zin A (1, 2 ) ⎤ ⎡ 1 ⎤ ( e ) ⎡ A (1, 2 ) ⎤ qout ⎢ ⎥ = qin ⎢ ⎥+P ⎢ ⎥ ⎣Zout ⎦ ⎣ A ( 2,1) + Zin A ( 2, 2 ) ⎦ ⎣ A ( 2, 2 ) ⎦
[4.177]
312
Fluid-structure interaction
The system [4.177] is solved by performing a few standard manipulations and the final result is written as: p ( L) =
Zout ( A ( 2, 2 ) A (1,1) − A ( 2,1) A (1, 2 ) ) P (
e)
Zout ( A (1,1) + Zin A (1, 2 ) ) − ( A ( 2,1) + Zin A ( 2, 2 ) )
[4.178]
q ( L ) = p ( L ) / Zout
If applied to a uniform tube, formula [4.178] gives: p ( L; k ) = + P ( q ( L; k ) =
e)
sin αout cos αin sin ( kL + αin + αout )
e p ( L; k ) P ( ) cos αout cos αin = iZ tan αout iZ sin ( kL + αin + αout )
[4.179]
Particularizing further the problem to a circuit open at the inlet and stopped at the outlet: αin = iε1 α out =
π + iε 2 2
[4.180]
where a small amount of dissipation is provided by the coefficients ε1 and ε 2 which are assumed to be much less than unity. Substituting the impedances [4.180] into [4.179], we obtain: p ( L; ω ) = + P (
e)
cos ( iε1 ) cos ( iε 2 ) ⎛ ωL ⎞ cos ⎜ + i ( ε1 + ε 2 ) ⎟ ⎜c ⎟ ⎝ f ⎠
[4.181]
Expanding [4.181] to the first order of the damping coefficients, the pressure response is written in the final form: p ( L; ω ) =
+ P( ) ⎛ ωL ⎞ +i ε +ε cos ⎜ ⎜ c ⎟⎟ ( 1 2 ) ⎝ f ⎠ e
[4.182]
Starting from this configuration of reference, the circuit is modified by inserting a cylindrical cavity of volume VE and radius RE much smaller than L as shown in
Plane acoustical waves in pipe systems
313
Figure 4.40. Large cavity inserted at some place in the uniform pipe
Figure 4.40. To derive the new response, it suffices to determine the transfer matrix of the modified circuit and then to apply the formula [4.178]. The modified matrix is:
[ Am ] = [ Aout ][ AE ][ Ain ]
[4.183]
where [ Ain ] refers to the tube element at the inlet (length L1 ), [ AE ] to the cavity and [ Aout ] to the tube element at the outlet (length L2 ):
−i ⎡ ⎤ ⎡ cos kL1 sin kL1 ⎥ cos kL2 ⎢ ; [ Aout ] = ⎢ [ Ain ] = ⎢ Z ⎥ ⎢ ⎢⎣ −iZ sin kL1 cos kL1 ⎥⎦ ⎢⎣ −iZ sin kL2 −i ⎤ ⎡ 1 ωV 1 ⎢ [ AE ] = ⎢ ZE ⎥⎥ where = E2 ZE ρ f c f ⎣⎢0 1 ⎦⎥
−i ⎤ sin kL2 ⎥ Z ⎥ cos kL2 ⎥⎦
[4.184]
After a few manipulations the following coefficients are obtained: Am (1,1) = cos kL −
Z sin kL1 cos kL2 ; ZE
Am (1, 2) =
⎛Z ⎞ Am (2,1) = iZ ⎜ sin kL1 sin kL2 − sin kL ⎟ ; ⎝ ZE ⎠
1 1 sin kL + cos kL1 cos kL2 iZ iZE
Am (2, 2) = cos kL −
Z cos kL1 sin kL2 ZE
[4.185] Substituting the coefficients [4.185] into [4.178], the response of the modified circuit is found to be:
314
Fluid-structure interaction
p ( L) =
cos αin sin αout P ( ) Z sin ( kL + αin + αout ) − sin ( kL1 + αin ) sin ( kL2 + αout ) ZE e
−i cos αout cos αin P ( ) q ( L) = ⎛ ⎞ Z Z ⎜ sin ( kL + αin + αout ) − sin ( kL1 + αin ) sin ( kL2 + αout ) ⎟ Z E ⎝ ⎠ e
[4.186]
The attenuation of the pressure response is conveniently measured by the ratio of the responses [4.186] on [4.179], which is written as: Ap ( L; k ) =
1 kV sin ( kL1 + α in ) sin ( kL2 + α out ) 1− E sin ( kL + αin + αout ) ST
[4.187]
ST denotes the cross-sectional area of the tube. The formula [4.187] points out the crucial importance of the ratio of the characteristic impedance of the tubes on that of the cavity to isolate the outlet tube from the source which excites the inlet tube. Furthermore, the ratio takes on the remarkably simple form, which indicates that the degree of isolation increases with the wave number, or frequency: Z kVE ωVE = = ZE ST c f ST
[4.188]
However, the degree of isolation does not improve in a monotonic way as frequency increases because of the modified resonances of the inlet and outlet tube elements. Such results are illustrated in Figure 4.41, where the values ε1 = ε 2 = 0.05 are assumed. The frequency of the fundamental in the initial configuration (VE = 0 ) is at 10 Hz. As VE increases, this mode tends to the Helmholtz mode of the inlet tube L1 = 11 m connected to the cavity. The second mode (third harmonic of the fundamental of the initial tube) tends to the first mode of the outlet tube, open at the cavity and stopped at the outlet. As is evident in the plot between 40 and 200 Hz, the higher modes are also deeply modified by the cavity, which acts essentially as a pressure node. Sound attenuation is very efficient except at a few resonances, especially in the low frequency range. Therefore, a large enclosure, or cavity, mounted in series in a circuit may be used to eliminate an embarrassing resonance, by locating the cavity near a pressure antinode of that resonance and to isolate the part of the circuit separated by the cavity from the acoustic source.
Plane acoustical waves in pipe systems
315
Figure 4.41. Pressure spectrum at the outlet induced by a unitary impulsive pressure source at the inlet, for three values of the volume ratio VE /VT
316
Fluid-structure interaction
4.3.3.2 Cavity connected in derivation to the main circuit
Figure 4.42. Cavity mounted in derivation on a pipe
Acoustical isolation can be performed also by mounting a large cavity in derivation on the pipe to be isolated, as shown in Figure 4.42. Actually, it is often much easier to achieve this kind of connection than to insert a large cavity in series in an already existing piping system. In this example, we are interested in determining the volume velocity at the outlet which is induced by a pressure source located at the inlet. To simplify the algebra of the problem, the terminal impedances are assumed to be conservative pressure nodes. On the other hand the cavity connected to the pipe can be modelled as a Helmholtz resonator of pulsation ω H . Due to the derivation, the present circuit differs qualitatively from those studied up to now, which comprised a single branch only. More generally, industrial and even domestic pipework systems incorporate multiple branches, often connected by large cavities or T-junctions. If the volume of the junction is small enough with respect to the wavelengths of interest, the conditions of connection are particularly simple, because they express the continuity of pressure and volume velocity. Extending the conditions [4.117] to the case of N tube elements connected to a junction device of negligible size, yields: p1 = p2 = .... = pN = pJ ; q1 + q2 + ....qN = 0
[4.189]
where p1 , p2 ,..., pN designate the fluctuating pressure at the interface between the tube element and the junction device and p J the pressure in the junction itself. q1 , q2 ,... are the volume velocities of the fluid flowing out from the tube elements into the junction.
Plane acoustical waves in pipe systems
317
Figure 4.43. Tube elements connected through a junction device of small size
Turning back to the pipe system of Figure 4.42, in the absence of the cavity, the volume velocity at the main tube outlet which is induced by the pressure source at the tube inlet is derived from the formula [4.178] as: q ( L; k ) =
P ( ) cos αin cos αout iZ sin ( kL + αin + αout ) e
[4.190]
In the presence of the cavity connected to the main tube at x = L1 , we first calculate the acoustic vectors at the junction as follows: ⎡ ⎡ q1 ⎤ ⎢ cos kL1 = q in ⎢p ⎥ ⎢ ⎣ 1⎦ ⎢⎣ -iZ sin kL1
1 ⎤ −i sin kL1 ⎤ ⎡ ⎢ (e) ⎥ ⎥ Z P ⎥ ⎥ ⎢Z + cos kL1 ⎥⎦ ⎢⎣ in qin ⎥⎦
[4.191]
Relation [4.191] is conveniently rewritten as: ⎡ q1 ⎤ qin ⎢ p ⎥ = cos α ⎣ 1⎦ in
⎡ sin kL1 ⎤ ⎡ cos ( kL1 + α in ) ⎤ (e) ⎢ + P iZ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ -iZ sin ( kL1 + α in )⎦ ⎣⎢ cos kL1 ⎦⎥
[4.192]
By using the transfer matrix [4.136] where the inertial coefficient is neglected, −ikVE ⎤ ⎡ ⎢1 ρ f cf ⎥ ⎡ 0 ⎤ ⎡ q2 ⎤ ⎢ ⎥ 2 ⎢ ⎢ ⎥= ⎥ ⎛ k ⎞ ⎥ ⎣ pH ⎦ ⎣ p2 ⎦ ⎢⎢ 0 1− ⎜ ⎟ ⎥ ⎢⎣ ⎝ k H ⎠ ⎥⎦
[4.193]
318
Fluid-structure interaction
Whence finally: ⎡ ⎡ 1 ⎤ ⎢ cos kL2 qout ⎢ ⎥ = ⎣Zout ⎦ ⎢⎢ -iZ sin kL 2 ⎣
−i sin kL2 ⎤ ⎥ ⎡ q3 ⎤ Z ⎥ ⎢⎣ p3 ⎥⎦ cos kL2 ⎥⎦
[4.194]
By using the conditions of continuity [4.189] at the junction, the relation [4.184] is transformed into: ⎡ ⎡ q1 + q2 ⎤ ⎢ cos kL2 = ⎢ ⎥ ⎣ p1 ⎦ ⎢⎢iZ sin kL 2 ⎣
i sin kL2 ⎤ ⎡ qout ⎤ qout = Z ⎥⎢ ⎥ ⎣ pout ⎦⎥ cos α out cos kL2 ⎦⎥
⎡ cos ( kL2 + α out ) ⎤ ⎢ ⎥ ⎣iZ sin ( kL2 + α out ) ⎦
[4.195]
Identifying the expressions of p1 arising from [4.195] and [4.192] respectively and then the expression of q2 arising from [4.195] and [4.193] respectively, one is led to solve a standard forced linear problem of the type: ⎡ B (1,1) B (1, 2 ) ⎤ ⎡ qin ⎤ P ( e ) ⎢ ⎥⎢ ⎥ = ⎣ B ( 2,1) B ( 2, 2 ) ⎦ ⎣ qout ⎦ iZ
⎡ F1 ⎤ ⎢F ⎥ ⎣ 2⎦
[4.196]
where the coefficients are found to be: B (1,1) = B (1, 2 ) =
sin ( kL1 + α in ) ; B ( 2, 2 ) = − cos ( kL2 + α out ) cos α in
sin ( kL2 + α out ) cos ( kL1 + α in ) − β H sin ( kL1 + α in ) ; B ( 2,1) = cos α out cos αin
F1 = cos kL1 ; F2 = − ( sin kL1 + β H cos kL1 ) βH =
[4.197]
kVE ⎛ ⎛ k ⎞2 ⎞ ST ⎜ 1 − ⎜ ⎟ ⎟ ⎜ ⎝ kH ⎠ ⎟ ⎝ ⎠
The solution can be expressed analytically in a simple compact form provided the main pipe is assumed to be terminated by two conservative pressure nodes. The result of practical interest is the attenuation factor for the volume velocity at the outlet: a ( βH ) =
qs ( β H ≠ 0 ) sin kL = qs ( β H = 0 ) sin kL − β H sin kL1 sin kL2
[4.198]
Plane acoustical waves in pipe systems
319
Figure 4.44. Attenuation coefficient provided by the Helmholtz resonator tuned to the fundamental of the main tube
Figure 4.45. Spectra of the volume velocity at the main tube outlet, excited by a unit impulsive pressure source at the inlet, please note that here q is expressed in cm 3 / s
Figure 4.44 displays the attenuation factor [4.198] for three values of the volume ratio VE /VT . The Helmholtz frequency f H = 20Hz is assumed to coincide with the
320
Fluid-structure interaction
fundamental of the main tube. The latter is the same as that shown in Figure 4.39, except that here the tube is open at both ends. The impedances are assumed to be αin = αout = 0.05i . In agreement with formula [4.198], the Helmholtz resonator is very efficient at attenuating the response in a rather broad frequency interval centred at the Helmholtz frequency. Outside this interval, the attenuation curve presents several peaks, some of them having a magnitude greater than unity, which means that at such frequencies the amplitude of the response is enhanced. However, to discuss the practical interest of the device in the whole frequency range, it is appropriate to complement the information displayed in Figure 4.44 with that of Figure 4.45, which displays the corresponding volume velocity spectra at the outlet induced by a unitary impulsive pressure source at the inlet. On such plots, it can be verified that the attenuation device is actually efficient in the whole range of frequencies, reducing significantly the level of the response spectra, except in a limited low frequency range, marked by the Helmholtz resonance of the modified system, combining the elasticity of the cavity and the inertia of the main tube. 4.3.4
Computational procedures suited to TMM softwares
4.3.4.1 Formulation of the forced acoustical system Generally, industrial and domestic piping systems, complicated as they may be, can still be viewed as composed of a single, or several distinct branches, terminated by impedances. In its turn, each branch is composed of several tube elements connected to each other by continuity conditions. A tube element is modelled by an elemental transfer matrix. At this step, several kinds of tube elements can be formulated analytically. Of course, the basic one is the uniform tube; however, conical tubes and even exponential and Bessel horn elements are also useful in many applications. In practice, the acoustical excitation is generally concentrated in a few particular locations within the circuit, where either a moving solid device, or a highly turbulent flow takes place. Typical examples of noisy mechanical devices are pumps, compressors, and fans, which act as periodic pressure and/or volume velocity sources, the period being related to the rotational speed of the device. Noise induced by turbulence of the flow conveyed by the pipe will be considered in Volume 4 of this series. Here, suffice it to say that the acoustical sources induced by turbulence are random in nature, comprising both a pressure and a volume velocity component, which can be characterized by their spectral properties. They extend over limited zones only, downstream geometrical accidents leading to flow singularities, like a sudden enlargement of the tube cross-section, an orifice, a T-junction, etc. All these sources can be conveniently included in a TMM model by prescribing at the appropriate location a finite discontinuity in pressure and/or volume velocity at the connection between two contiguous elements.
Plane acoustical waves in pipe systems
321
To extend the transfer matrix method to complicated circuits it would be inappropriate, and impracticable anyway, to use the same procedure as that presented in the last subsections, which consisted of reducing the whole circuit to a single 2 × 2 transfer matrix by eliminating all the dependent variables. Not only do the trigonometric manipulations involved in such an analytical approach become unduly tedious and intricate as the number of tube elements increases beyond two or three of them, but as a result of the process, the output is limited to the acoustic vectors at the terminations of the whole circuit. Consequently, to perform numerical calculations by using the computer, it is found suitable to reformulate the problem as an equivalent forced linear problem of the type: ⎡⎣ A ( k ) ⎤⎦ ⎡⎣U ( k ) ⎤⎦ = ⎡⎣ E ( k ) ⎤⎦
[ A]
[4.199]
is a regular matrix. [U ] denotes the vector of the unknowns and [ E ] that of
the external excitation. In the present problem, all these quantities depend on wave number k, or equivalently frequency f. Equation [4.199] is solved for a set of discrete frequency values, to produce the discrete Fourier transform (in short DFT) of the response to the acoustical sources, described by the DFT of the pressure and volume velocity jump signals. Concerning the DFT and numerical procedures to solve large linear algebraic systems, see for instance [AXI 04]. Since in [4.199] the intermediate and dependent variables are no longer eliminated, the acoustic response [U ] is made available at the ends of each tube element, as further described in the next subsections. 4.3.4.2 Matrix equation of a tube element The procedure to transform the matrix equation of a tube element, identified by its element number N, is described in Figure 4.46. On the left-hand side of the figure, the analytical formulation is recalled, where the acoustic vectors at the inlet and outlet of the element stand for the unknowns. The reason for not relating yet at this step the numbering of the unknowns to that of the elements will be made clear a little later. Turning to the computational model depicted on the right-hand side of the figure, as in the case of the meshes used in the finite element method, the tube element can be viewed as a line segment delimited by two nodes. However, similarity with the FEM ends here, because of the following differences which are worthwhile to be stressed again here. First the nodal unknown used in the TMM is mixing nodal displacement and stress variables. Consequently, as indicated in formula [4.199], the forced dynamical problem is discretized by using a single matrix instead of the stiffness and mass matrices used in the FEM. Secondly, the N elemental matrix ⎡⎣ A( ) ⎤⎦ is based on exact analytical solutions, the validity of which is independent of the element length and correlatively of the wavelength, at least within the range of validity of the plane wave approximation, and not based on
322
Fluid-structure interaction
approximated solutions of potential and kinetic energies using low degree polynomials.
Figure 4.46. Tube element
The necessary transformation of the analytical matrix equation to formulate the computational model in accordance with the forced problem [4.199] is also shown in Figure 4.46. As no acoustical source was supposed to excite the tube element, the excitation vector on the right-hand side of the equation is zero. On the other hand, a further transformation is necessary, which consists in defining the vector [U ] by using the components of the nodal acoustic vectors. Accordingly, the elemental equation is written as the two following lines in the assembled matrix: A11( )
A12( )
−1
( ) A21
( ) A22
0
N N
N N
qn pn 0 = −1 qn +1 0 pn +1 0
[4.200]
4.3.4.3 Impedances and external sources An impedance can be formulated as a homogeneous relationship between the pressure and the volume velocity at a given position, hence at a given node. This corresponds to an extra line in the matrix equation and to a single node element in the TMM mesh of the circuit, as shown in Figure 4.47. However, to account for the possibility that external sources are located at the inlet and the outlet of the circuit, it is appropriate to transform further the computational model by using a fictitious two nodes element, termed connection element, which serves here to connect the impedance element to the tube element. As shown in Figure 4.48, the connection element does not introduce any supplementary nodal variable, but simply accounts for the continuity conditions to be satisfied either in the absence or in the presence of external acoustical sources between two physical elements.
Plane acoustical waves in pipe systems
323
Figure 4.47. Terminal impedances modelled as one node connecting element
Figure 4.48. Terminal impedances modelled as two nodes connecting element
Considering for instance a circuit comprising the inlet and outlet impedances, a pressure and volume velocity source located at the inlet and a single tube element, the corresponding assembled matrix equation is written as:
324
Fluid-structure interaction
⎡Zin ⎢1 ⎢ ⎢0 ⎢ ⎢0 ⎢ ⎢0 ⎢0 ⎢ ⎢0 ⎢⎢ 0 ⎣
−1 0 1
0 −1 0
0 0 −1
0 0 0
0 0 0
0 0 0
0
A11( )
A12( ) 1
−1
0
0
0
(1)
A21
(1)
A22
0
−1
0 0 0
0 0 0
0 0 0
1 0 0
0 1 0
0 −1
1
0 Zout
0 ⎤ ⎡ q1 ⎤ ⎡ 0 ⎤ 0 ⎥ ⎢ p1 ⎥ ⎢ −Q ( e ) ⎥ ⎥⎢ ⎥ ⎢ ⎥ 0 ⎥ ⎢ q2 ⎥ ⎢ − P ( e ) ⎥ ⎥⎢ ⎥ ⎢ ⎥ 0 ⎥ ⎢ p2 ⎥ ⎢ 0 ⎥ =⎢ ⎥ ⎥ 0 ⎥ ⎢ q3 ⎥ ⎢ 0 ⎥ ⎢ ⎥ 0 ⎥ ⎢ p3 ⎥ ⎢ 0 ⎥ ⎥ ⎢ ⎥ −1⎥ ⎢⎢ q4 ⎥⎥ ⎢ 0 ⎥ −1⎥⎦⎥ ⎣⎢ p4 ⎦⎥ ⎢⎣⎢ 0 ⎥⎦⎥
[4.201]
4.3.4.4 Single branched circuits
Figure 4.49. Single branched circuit comprising two tube elements
A single branched circuit comprising several tube elements is modelled in the same way as the single tube element presented just above, by using the connecting element to connect the contiguous tube elements. Let us consider for instance a circuit comprising two distinct tube elements excited by a pressure source located at the interface between the two tubes, see Figure 4.49. The corresponding assembled matrix equation is written as:
Plane acoustical waves in pipe systems
⎡ Zin ⎢ 1 ⎢ ⎢ 0 ⎢ ⎢ 0 ⎢ ⎢ 0 ⎢ 0 ⎢ ⎢ 0 ⎢ ⎢ 0 ⎢ 0 ⎢ ⎢ 0 ⎢ 0 ⎢ ⎣⎢ 0
−1 0 1
0 −1 0
0 0 −1
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
0
A11( )
A12( ) 1
−1
0
0
0
0
0
0
0 0 0
(1)
A21 0 0
(1)
A22 0 0
0 1 0
−1 0 1
0 −1 0
0 0 −1
0 0 0
0 0 0
0 0 0
0
0
0
0
0
A11( )
A12( ) 2
−1
0
0
0 0 0 0
( 2)
( 2)
0 1 0 0
−1 0 1 0
0 −1 0
0 0 0 0
1
0 0 0 0
0 0 0 0
0 0 0 0
2
A21 0 0 0
A22 0 0 0
Zout
325
0 0 0
⎤⎡q ⎤ ⎡ 0 ⎤ ⎥⎢ 1⎥ ⎢ ⎥ ⎥ ⎢ p1 ⎥ ⎢ 0 ⎥ ⎥ ⎢q ⎥ ⎢ 0 ⎥ ⎥ 2 0 ⎥ ⎢ p2 ⎥ ⎢ 0 ⎥ ⎢ ⎥ ⎥⎢ ⎥ 0 ⎥ ⎢ q3 ⎥ ⎢ 0 ⎥ ⎥ ⎢ ⎥ ⎢ 0 ⎥ ⎢ p3 ⎥ ⎢ 0 ⎥ ⎥ ⎢ ⎥ = ⎢ (e) ⎥ 0 ⎥ q4 ⎢−P ⎥ ⎥ ⎢⎢ p ⎥⎥ ⎢ 0 ⎥ 0 ⎥ 4 ⎥ ⎢ ⎥ ⎢ 0 ⎥ ⎢ q5 ⎥ ⎢ 0 ⎥ ⎥ ⎢ ⎥ 0 ⎥ ⎢ p5 ⎥ ⎢ 0 ⎥ ⎢q ⎥ ⎥ 0 ⎥ −1 ⎢ 6 ⎥ ⎢ ⎥ ⎢⎢ 0 ⎥⎥ p ⎢ ⎥ 6 ⎣ ⎦ ⎣ ⎦ −1 ⎥⎦
[4.202] 4.3.4.5 Multi-branched circuits To deal with multi-branched pipework systems, each branch is modelled as explained in the last subsection and then a procedure is devised to connect the branches together. Here, for the sake of simplicity, presentation is restricted to junction devices of small size, in such a way that the connection conditions [4.189] hold. A convenient manner for assembling the different branches of a circuit is to devise a single node junction element, which is associated with a single new variable, namely the pressure within the junction. The assembling procedure, particularized to the case of a single junction element, is depicted in the logic chart of Figure 4.50. The junction element is denoted (J) and the fluctuating pressure within it is pJ . First, the branches are identified as those parts of the circuit which are terminated by an impedance and the junction. By convention, the impedance termination is taken as the inlet of the branch and the junction as the outlet, in such a way that a positive volume velocity in each connected branch means a flow entering into the junction. As indicated schematically by the logical box, in the assembled linear system of equations, each branch is described according to the procedure presented in the last subsection and is terminated by a connecting row which states that the pressure at the branch outlet is equal to the pressure pJ within the junction. Finally, the last row of the system corresponds to the single node junction element, stating that mass flow through the junction device is zero, to comply with the massconservation law. However, so long as the fluid is homogeneous, the mass flow can be replaced by the volume velocity.
326
Fluid-structure interaction
Figure 4.50. Logic chart for connecting the branches of a piping system
Figure 4.51. Connection of three branches to a junction element
The procedure is illustrated in Figure 4.51, taking the example of a circuit comprising three branches terminated by the impedances Z1, Z2 , Z3 and connected to a junction (node J) at the nodes l+1, m+1, n+1, respectively. However, to illustrate the
Plane acoustical waves in pipe systems
327
way the matrix is assembled, we consider a circuit restricted to two branches, of a single tube element each, connected by a junction. The sole interest of this example is to allow us to write down a matrix equation of manageable size, which is presented here as: ⎡Z1 ⎢1 ⎢ ⎢0 ⎢ ⎢0 ⎢ ⎢0 ⎢0 ⎢ ⎢0 ⎢0 ⎢ ⎢0 ⎢ ⎢0 ⎢0 ⎢ ⎢0 ⎢ ⎣⎢ 0
-1 0
0 −1
0 0 −1
0 0
0 0
0 0
0 0
0 0
0 0
1
0
0
(1)
A11
(1)
0
0
0
0
0
0
A12
0 0 0 0 0
(1)
A21
(1)
−1
0
0
0
0
0
A22
0 0 0 0
0 0 0 0 0
−1
0 0 0 0
0 0 Z2 1 0
0 0 −1 0 1
0 0 0 −1 0
0 0 0 0 −1
0
0
0
0
0
0
0
A11( )
A12( )
0 0 0
0 0 0
0 0 0
0 0 1
0 0 0
0 0
0 0
( ) A21
( ) A22
0
0
0
0
0
0
1 0 0 0
2 2
2 2
0 0
0 0
0⎤ q ⎡ 1⎤ ⎡ 0 ⎤ 0 ⎥⎢ p ⎥ ⎢ 0 ⎥ ⎥⎢ 1⎥ ⎢ ⎥ 0 0 0 ⎥⎢q ⎥ ⎢ 0 ⎥ ⎥ 2 ⎢ ⎥ 0 0 0 ⎥ ⎢ p2 ⎥ ⎢ 0 ⎥ ⎢ ⎥ ⎥ 0 0 0 ⎥ ⎢ q3 ⎥ ⎢ 0 ⎥ ⎥ ⎢ ⎥ ⎢ 0 0 −1⎥ ⎢ p3 ⎥ ⎢ 0 ⎥ ⎥ 0 0 0 ⎥ ⎢ q4 ⎥ = ⎢⎢ 0 ⎥⎥ ⎢ ⎥ e 0 0 0 ⎥ ⎢ p4 ⎥ ⎢ −Q ( ) ⎥ ⎥⎢ ⎥ ⎢ (e) ⎥ 0 0 0 ⎥ q5 ⎢−P ⎥ ⎢ ⎥ ⎥ −1 0 0 ⎥ ⎢ p5 ⎥ ⎢ 0 ⎥ ⎥ ⎢ ⎥ ⎢ 0 −1 0 ⎥ ⎢ q6 ⎥ ⎢ 0 ⎥ ⎥⎢ ⎥ ⎢ ⎥ 0 1 −1⎥ ⎢ p6 ⎥ ⎢ 0 ⎥ ⎥ 1 0 0 ⎦⎥ ⎣⎢ pJ ⎦⎥ ⎢⎣ 0 ⎥⎦
[4.203] 4.3.4.6 Application to the acoustical isolation of a forced flow loop As an application, the circuit sketched in Figure 4.52(a) is considered. It stands for a water-flow loop used as a test rig to measure the noise produced by various obstacles. The flow is forced into the loop through an organ which is also noisy, and modelled here as a broad banded pressure source, see Figure 4.52. Acoustic impedance at the source is assumed to be a dissipative pressure node. As shown in Figure 4.51(b), the loop is modelled as a uniform tube (inner radius R = 0.3m) terminated by equal dimensionless impedances ( αin = αout = 0.04π i ). Finally, as shown in Figure 4.52(c), the central part of the pipe, where the obstacles are inserted and acoustic responses measured, is isolated from the undesirable source by connecting it at the inlet and the outlet by two identical tanks of large volume, which delimit thus the test section. The efficiency of the tanks to isolate the test section is illustrated in Figure 4.53, where the spectral density of the pressure response at the middle of the central part is plotted for three values of the volume of the tanks.
328
Fluid-structure interaction
Figure 4.52. Acoustical isolation of a test rig by insertion of large enclosures
The results displayed in Figure 4.54 can be qualitatively understood based on those already discussed in subsection 4.3.3.1. To obtain a high degree of isolation it is necessary that the volume of the enclosures becomes much larger than that of the test section (about 1.13 m 3 ). On the other hand, due to the enclosures, there is a low frequency mode, interpreted as the Helmholtz resonance of the central part of the circuit. Elasticity of this mode is provided by the enclosures and inertia by the water column within the test section. Similar results would be obtained by assuming a volume velocity source instead of a pressure source associated to a volume velocity node: α in = α out =
π + iε 2
Plane acoustical waves in pipe systems
329
Figure 4.53. Spectral density of the pressure source
Figure 4.54. Spectral density of the pressure at the middle part of the test section for three values of the enclosure volume
4.4. Speed of sound 4.4.1
Speed of sound and fluid compressibility
The speed of sound and Young’s modulus as defined by relations [4.2] and [4.3] can be considered as mechanical properties related to fluid compressibility. In this
330
Fluid-structure interaction
context, it is also of interest to examine further the analogy with the longitudinal waves travelling along a chain of a large number N of identical mass-spring systems, which was already studied in [AXI 04] to illustrate the concept of travelling and standing waves. Actually, such a chain corresponds to the “sonorousline” model used in early theories of sound propagation, by Lagrange in particular, as mentioned for instance in [PIE 91]. As seen in Figure 4.55, a mass-spring subsystem of the chain is characterized by the length L0 and the mass and stiffness coefficients K 0 and M 0 , respectively. The chain as a whole is characterized by the length L = NL0 , the stiffness coefficient K = K 0 / N and the mass M = NM 0 . The magnitude of the force required to contract or expand the chain by the length X is F = KX .
Figure 4.55. Discrete model of the oscillations of material points in longitudinal waves
Accordingly, a coefficient of compressibility of the chain may be defined as: κ=
X 1 N 1 = = = LF LK LK 0 L0 K 0
[4.204]
On the other hand the density of the chain is given by: ρ=
M M0 = L L0
[4.205]
Substituting [4.204] and [4.205] into [4.2], the following wave speed is obtained:
Plane acoustical waves in pipe systems
c=L
K K0 = L0 M M0
331
[4.206]
As a particularly interesting point, the result [4.206] is found to be independent of the number of oscillators. However, the actual wave speed in a finite chain differs from this simple result, as already pointed out in [AXI 04] Chapters 6 and 8, where the dispersive nature of propagation was emphasized. In fact, starting from the modal quantities for a chain fixed at each end, it is not difficult to verify that the result [4.206] is in agreement with the asymptotic form of the natural pulsations: nπ ⎛ ⎛ nπ ⎞ ⎞ K 0 ω n = ⎜ 2sin ⎜ ; lim ω = nω1 = ⎟⎟ n 2 N M N ⎝ ⎠⎠ ⎝ 0 N →∞ ⎛ nπ x ⎞ ϕ n ( x ) = sin ⎜ ⎟ ⎝ L ⎠
K0 M0
[4.207]
where the results [4.207] stand for the formulas [6.37] and [8.14] established in [AXI 04] and repeated here for convenience. The phase speed of the wave is now defined as: cψ =
ω ω n ⎛ nπ = =⎜ k kn ⎝ N
K0 ⎞ ⎛ L ⎞ L ⎟⎜ ⎟= M 0 ⎠ ⎝ nπ ⎠ N
K0 K0 K = L0 =L M0 M0 M
[4.208]
Here k is the wave number and kn = nπ / L its modal value. The phase velocity [4.208] tends to the asymptotic expression [4.206] in the low frequency range ω << ω 0 = K 0 / M 0 . Incidentally, it may be worth mentioning that concerning the natural modes of vibration in a crystal lattice, termed phonons, a clear distinction is made between the acoustic phonons which correspond to the low frequency and large wavelength range ω << ω 0 ; λ >> L0 and the optical phonons which correspond to the high frequency and short wavelength range. On the other hand, the sonorous line model provides us with a qualitative explanation of the large decrease observed in the values of the Young’s modulus when matter is passing from the solid to the liquid state and then from the liquid to the gaseous state. In solids, the particles (atoms, or molecules) are so tightly packed that the interaction forces acting at short distance dominate both the static equilibrium position of the particles and the restoring forces that tend to bring the particles back to it. The motion due to thermal vibration is usually negligible with regard to the mechanical vibrations of interest. As a consequence, the speed of sound is only poorly dependent on temperature. At the extreme opposite, in gases the spacing between particles is large enough so that inter-atomic or intermolecular forces are very small and in fact are entirely discarded in the kinetic theory of perfect gases. On this subject, the reader can be referred to many textbooks, in particular to [MOO 55], [FEY 63] and [KAU 66]. According to the kinetic theory,
332
Fluid-structure interaction
in a perfect gas macroscopically at rest, the motion of the particles is dominated by the thermal component. The latter is entirely governed by temperature which determines the amount of kinetic energy imparted to the gas and the number of elastic collisions between particles, which occur at random. Accordingly, the model of the well ordered chain of linear oscillators is no more valid. Nevertheless, it can be suitably replaced by a collection of “impact” oscillators characterized by an impact stiffness coefficient much larger than the linear stiffness coefficient, as already described in [AXI 04] Chapter 5 and [AXI 05] Chapter 4. Such a collection comprises a huge number of oscillators acting indifferently in any direction. In a macroscopically quiescent and isothermal fluid, the resulting “molecular motion” is an isotropic random walk. The mean free path of the particles between two successive collisions is found to be: =
V V = 2 2π d N 2π d 2 nN A
[4.209]
N denotes the number of particles contained in the volume V and d their diameter. In air at STP conditions, is about 10−4 mm . The number of moles is n and N A = 6.0221023 is the Avogadro number which specifies the number of particles in a mole.
The pressure can be connected directly to the resulting change of momentum of the impacting particles. This can be demonstrated by considering a portion of gas enclosed in a cubic box (length L) provided with rigid walls. Assuming monoatomic particles, for the sake of simplicity, kinetic energy is connected to the motion of the mass centre of the particle (translation) only and is expressed as: Eκ =
2 Nm0 vrms 2
[4.210]
where m0 is the mass of a particle and vrms = v 2 is the root mean square (in short r.m.s.) velocity of the particles, expressed in terms of the Cartesian components of velocity as: 2 vrms = v x2 + v 2y + v z2
[4.211]
Since the thermal motion is isotropic, r.m.s. velocities are the same in any direction, so the following relation holds: 2 vrms = 3v x2 = 3v 2y = 3v z2
[4.212]
On the other hand, considering first a single particle moving along the direction Ox and impacting a fixed wall normal to Ox, the mean change in momentum is:
Plane acoustical waves in pipe systems
2 δp=− m0 vrms i 3
where, as an average, the particle velocity at incidence is
333
[4.213] v x2 i = vrms / 3 . The
mean number of collisions per second is: Nc =
Nvrms 3L
[4.214]
Then, the change of momentum per second, which is equivalent to the impacting force of the gas exerted on the wall is found to be: 2 δ P 2m0 Nvrms i [4.215] F =− = δt 3L The pressure exerted on the walls follows as: P=
2 F F m Nv 2 Nm0 vrms = 2 = 0 3rms = 3L 3V S 2L
[4.216]
The relation [4.216] is finally written as: PV =
2 Nm0 vrms 2 = Eκ 3 3
[4.217]
Identifying the material law [4.217] inferred from the kinetic theory with that derived from experiment, temperature is conceptually connected to kinetic energy of thermal motion: 2 Nm0 vrms 2 = Eκ = nR T 3 3
[4.218]
The r.m.s. velocity of the particles is found to be: vrms =
3nR T 3N R T 3R T = = m0 N m0 NN A M
[4.219]
Comparing [4.219] and [1.33] it can be concluded that the r.m.s. velocity of the gas particles is of the same order of magnitude as the speed of sound in the gas continuum. The liquid state can be qualitatively understood as an intermediate state between a crystalline solid and a perfect gas, in which the amplitude of the thermal motion of particles is of the same order of magnitude as the mean distance between the particles. Therefore, there is still some order in the molecular structure.
334
Fluid-structure interaction
4.4.2
Isothermal versus adiabatic speed of sound in gases
Up to here, presentation was focused on the mechanical aspect of the subject without addressing the problem of the thermodynamic changes which can occur within the fluid as a sound wave passes by. As already seen in Chapter 1, such changes can be accounted for in a gas by introducing the polytropic law repeated here as: Pρ
−γ p
−γ p
= P0 ρ 0
[4.220]
The polytropic index γ p is equal to one if the transformation is isothermal, and takes on the adiabatic value γ = C p / CV if no heat is transferred (see Appendix A1). The speed of sound in a perfect gas is given by formula [1.33], repeated here as: c0 = γ p
γ pR P0 = T0 M ρ0
[4.221]
As in Chapter 1, the subscript ( 0 ) is used here to mark that the value of the related quantity refers to the static state of equilibrium. The problem is thus to determine the value of γ p which is appropriate to fit with reality. Newton’s prediction in Book II of the Principia (1687) was in agreement with the isothermal model. However, measurements of the speed of sound made in air since the seventeen century (on the first measurements by Marin Mersenne published in 1635 and 1644, see [LEN 51,52]) demonstrated that the isothermal value was too small by about 16%. Laplace was the first who understood that sound waves in gases are in fact adiabatic rather than isothermal. Accordingly, the quantity which is invariant in the gas as a sound wave passes by is the heat (or entropy) and not the temperature. Of course, in Newton’s days the concepts of isothermal and adiabatic processes were not known. To understand the reason why the adiabatic instead of the isothermal law holds, let us consider a pressure wave of the progressive and harmonic type p exp ( iω ( t − x / c0 ) ) . If the transformation is adiabatic, temperature is expected to increase by a small amount at the pressure crests and to decrease at the troughs, see Appendix A1. Accordingly, we could expect a wavy temperature profile of the kind: θ = θ 0 exp ( iω ( t − x / c0 ) )
[4.222]
Nevertheless, due to the temperature gradient some heat must be transferred irreversibly from the “hot” (crests) to the “cold” (troughs) regions. Such a mechanism is termed thermal conduction. Due to its irreversible nature, the process is dissipative and tends to make the temperature field uniform. Thus, the basic idea
Plane acoustical waves in pipe systems
335
is that the adiabatic law prevails in sound propagation because thermal conduction is too slow a process for heat adjustment to have time to occur as the sound wave passes by. Accordingly, gas dilatation in small amplitude sound waves can be modelled as an adiabatic and reversible process, which means an isentropic process. Heat conduction is governed by the Fourier law: jH = −κ H grad T [4.223] where jH is the heat vector flux and κ H the thermal conductivity of the material. The value of κ H varies by a large amount depending on the material. For instance in air at STP conditions κ H = 0.026 W/m°K whereas in copper κ H = 401 W/m°K . However, what matters in the present problem is not so much the amount of heat transferred, but the change in temperature induced by conduction. The amount of heat Q transferred by conduction to some place can be inferred from the Fourier law by using the divergence theorem, which yields the heat transfer equation: DQ DQ − div jH = 0 ⇔ − κ H ΔT = 0 Dt Dt
[4.224]
Heat is related to temperature by using the specific heat coefficient at constant pressure (state of reference P0 ), as follows: Q = ρ 0C P T
[4.225]
Substituting [4.225] into [4.224], the following equation is obtained: DT − χ H ΔT = 0 Dt
[4.226]
Equation [4.226] is known as the heat diffusion equation and χ H is the thermal diffusivity, defined as: χH =
κH ρ 0C P
[4.227]
The characteristic length of diffusive transport of heat is termed the diffusion length and denoted RDL . Using the heat diffusion equation it is found that: RDL = τχ L = χ L / ω
[4.228]
where τ is the scale factor for the time change of temperature, taken as ω −1 in the case of a harmonic wave. To assess the importance of heat diffusion in sound waves, we substitute the temperature profile [4.222] into [4.226], which yields:
336
Fluid-structure interaction 2
⎛ω ⎞ iω − χ H ⎜ ⎟ = 0 ⇔ ic0 − χ H k = 0 ⎝ c0 ⎠
[4.229]
As a consequence, temperature change due to heat conduction is found to be negligible if the following inequality holds: χ H k << c0
[4.230]
Using the definition [4.228], the condition [4.230] can be expressed in terms of the diffusion length and the wavelength as: RDL << 1 λ
[4.231]
In a gas, using the kinetic theory, it can be shown that: χ H ≈ c0
[4.232]
where is the mean free path of the molecules. The reasoning to establish [4.232] is worthwhile outlining here because it is of great help for the general understanding of the diffusive transport processes. Let us consider two adjacent layers of fluid at distinct temperatures T1 and T2 , by convention T1 is assumed to be larger than T2 . The particles initially in the T1 region are termed “hot”, whereas the particles initially in the T2 region are termed “cold”. Due to the thermal random walk, particles are continually jumping from one layer to the other. At each time a collision occurs some amount of heat energy is transferred from a “hot” to a “cold” particle. On the macroscopic scale, the process continues so long as the temperatures are not equalized. Assuming the temperature gradient is along the direction Oz, the thermal energy transferred during one collision can be written as: δQ =
∂Q ∂z
[4.233]
The mean number of collisions per second and per unit area is: nc =
Nvrms nvrms = 3V 3
[4.234]
where n is the number of particles per unit volume and nc the mean number of particles per unit area travelling in the Oz direction.
Plane acoustical waves in pipe systems
337
Table 4.1. Thermal properties of a number of materials Material
Density
Kg/m 3
Heat capacity
Thermal conductivity
Thermal diffusivity
106 J/m 3
W/mK
10−8 m 2 / s
Air
1.19
0.001
0.025
1938
Water
1000
4.18
0.6
14
Ice
917
2.017
2.1
104
Alcohol
800
2.43
0.17
7
Aluminium
2700
2.376
237
9975
Copper
8960
3.494
390
11161
Stainless steel
7900
3.95
16
405
Concrete
2200
1.94
1.28
66
Marble
2700
2.376
3
126
Glass
2600
2.184
0.93
43
Foam glass
120
0.092
0.04
49
Foam glass
120
0.092
0.04
49
Hence the heat flux is: jH =
nvrms ∂Q nvrms ∂Q ∂T ∂T = = κH ∂z 3 ∂z 3 ∂T ∂z
[4.235]
Finally, using the relation [4.225] the thermal conduction is written as: κH =
nvrms ∂Q nm0CP vrms ρ 0CP vrms ρ 0 c0CP = = 3 ∂T 3 3 3
[4.236]
338
Fluid-structure interaction
The thermal diffusivity follows as: χH =
κH c 0 ρ 0C P 3
[4.237]
and the inequality [4.230] reads also as: 2 << 1 λ
[4.238]
As stated before, in air and at STP, = 0.1 μm while the wavelength of a sound wave at a frequency of 10 kHz is about 3 cm. Hence the condition to discard thermal conduction is more than adequately fulfilled. By using tabulated values of thermodynamic properties of liquids and solids, it can be verified that the condition [4.230] holds in most, if not all, monophasic fluids, gases or liquids (see for instance the indicative table 4.1) . In air γ 1.4 and the adiabatic value of speed of sound is about 1.18 times the isothermal value, as observed. Substituting the molar mass of air M = 2 910−3 kg , into [4.221] yields c0 20.1 T0 . At 20° C, in air c0 342 m/s . In liquids, there is a lack of analytical formulation of the state equation. So, speed of sound amongst other physical properties, is described by using empirical or semi-empirical relationships. For instance, the speed of sound in water at atmospheric pressure can be related to the temperature θ expressed in Celsius degrees, through the following empirical formula: cL ≅ 1403 + 5θ − 0.06θ 2 + 0.003θ 3 m/s
[4.239]
As quoted from [PIE 91], the earliest measurements of speed of sound in water were made by J.D. Colladon at Lake Geneva in 1826; the result was cw = 1435m / s at θ = 8°C . The present accepted value is cw = 1439.1m / s , so very close to that by Colladon. Comparison of [4.221] and [4.239], indicates that speed of sound in a gas is much more temperature dependent than in a liquid. Temperature changes connected to a sound wave are also much less marked in a liquid or solid than in a gas, since heat capacity is much higher. Finally, the plots presented in Figure 4.56 using water and steam tables published in [SCH 81] indicate that sound velocity is poorly dependent on pressure, whereas a very significant decrease occurs in the high temperature range for pressurized water.
Plane acoustical waves in pipe systems
339
Figure 4.56. Sound velocity in water as a function of pressure and temperature
4.4.3
Speed of sound in a gas liquid mixture (bubbly liquid)
4.4.3.1 Quasi-static homogeneous model Let us consider a column of “bubbly fluid” which is a mixture of liquid water interspersed with bubbles of an incondensable gas, for instance air, see Figure 4.57. The density and speed of sound in the liquid are denoted ρ L and c L and in the gas ρG and cG . The fraction of volume occupied by the gas is termed void fraction and defined here as: αH =
VG V = G VG +VL VH
[4.240]
340
Fluid-structure interaction
Figure 4.57. Liquid column containing gas bubbles
VG is the volume occupied by the gas and VL that occupied by the liquid. Hence, α H varies from 0 to 1 when passing from a pure liquid to a pure gaseous column of
fluid. The subscript
(H )
stands for “homogeneous”. According to the
homogeneous model, which is the simplest conceivable, the distribution and the size of the bubbles are assumed to be uniform, which means in fact that the actual properties of the individual bubbles are averaged on the column volume. We are interested here to determine the inertial and elastic properties of the homogeneous mixture in terms of the properties of the liquid and gaseous phases, as functions of the homogeneous void fraction. Determining the density ρ H of the mixture is immediate using the principle of mass conservation: ρ HVH = ρGVG + ρ LVL ⇒ ρ H = ρGα H + ρ L (1 − α H )
[4.241]
Concerning the elastic behaviour of the mixture, a short cut reasoning is to claim that when the column is compressed, or expanded, the gas and the liquid act as two springs connected in series. On the other hand the compressibility of each phase is proportional to the fraction of volume it occupies in the mixture. Accordingly, the equivalent stiffness coefficient of the column can be written as: 1 1 1 1 1 − αH αH = + ⇔ = + K H K L KG EH EL EG
Whence, the Young’s modulus of the homogeneous mixture:
[4.242]
Plane acoustical waves in pipe systems
⎛ α 1 − αH ⎞ E H = ρ H cH2 = ⎜ H 2 + ⎟ ρ ρ L cL2 ⎠ c ⎝ G G
341
−1
[4.243]
and the speed of sound: cH =
ρ EH ⎛ 1 ⎛ 2 = ⎜ 2 ⎜ α H + α H (1 − α H ) L ρ H ⎝ cG ⎝ ρG
⎞ 1⎛ ρG 2 ⎟ + 2 ⎜ (1 − α H ) + α H (1 − α H ) ρL ⎠ cL ⎝
⎞⎞ ⎟⎟ ⎠⎠
−1/ 2
[4.244] It is not difficult to check the validity of such reasoning by studying in detail the compression of the fluid column by a small amount δ p . In the quasi-static range, corresponding to the low frequency domain much smaller than the resonance frequencies of the bubbles, (cf. Chapter 3 subsection 3.3.2.3) the pressure transmitted to the mixture and to the individual phases can be assumed to be the same, because the inertia of the liquid is neglected. Hence, δ p verifies the following relationships: δ p = − EH
δVH δV δV = − EG G = − E L L VH VG VL
[4.245]
On the other hand, the change in volume is: δVH = δVG + δVL
[4.246]
and with the aid of [4.245], we arrive at: δVH = δVG + δVL = −
⎛V VH V ⎞ δ p = − ⎜ G + L ⎟δ p EH E ⎝ G EL ⎠
[4.247]
Using [4.240], relation [4.243] is easily recovered. Figure 4.58 displays ρ H and E H as functions of α H , for a mixture of air and water at STP ( P0 = 10 5 Pa, θ = 20°C ). As could be expected, the density and Young’s modulus are monotonic decreasing functions of 1- α H , varying from the value in pure liquid to that in pure gas. However, the density takes on values close to that of water up to void fractions of a few tenths. On the other hand, a tiny amount of gas is enough to induce a substantial decrease in E H . This is clearly a consequence of the large values of the liquid to gas ratios of density and Young’s modulus.
342
Fluid-structure interaction
Figure 4.58. Density and Young’s modulus in the air-water homogeneous mixture
In Figure 4.59, c H is plotted versus α H . Though often perceived as less intuitive, the result that speed of sound in the mixture can be much lower than in each individual phase taken separately, is in fact natural since, in a large range of α H values, the density of the mixture is not far from that of the liquid while the
Plane acoustical waves in pipe systems
343
Figure 4.59. Speed of sound in the air-water homogeneous mixture
compressibility is not far from that of the gas. So, as soon as a tiny amount of gas is present in the mixture, c H = E H / ρ H is substantially decreased. A minimum of about 26 m/s occurs at α H = 0.5 . To obtain values not far from cL , α H must be less than a few 10−5 and to obtain values not far from cG , α H must be higher than about 0.995. Though the homogeneous model is clearly unable to account for all the
344
Fluid-structure interaction
aspects of sound propagation in two-phase flows, it was used successfully to interpret laboratory experiments on plane acoustical waves in pipe systems conveying cavitating flow, as reported for instance in [AXI 91]. Another interesting point to note is that the gas phase of the mixture behave isothermally, as studied in detail and from several distinct standpoints in [PLE 60] and [HSI 61]. Restricting here the presentation to the qualitative reasoning given in [PLE 60], it can be shown that the temperature change within a gas bubble due to the compression or the rarefaction of the wave is largely prevented by conduction of heat to the liquid. Considering a bubble of mean radius R0 , the increase in thermal energy connected to an increment in temperature ΔT is: 4 ΔQG = π R03 ρG CG ΔT 3
[4.248]
CG is the specific heat per unit mass of the gas.
On the other hand, the mean flow of heat from the bubble to the liquid phase due to conduction is approximately: dQ ΔT κL ( 4π R02 ) dt RDL
[4.249]
where κ L is the thermal conduction coefficient of the liquid. Integrating the heat flow [4.249] for a characteristic time π / ω , we obtain the heat transferred from the bubble to the liquid during the compression half-cycle: Q κL
ΔT π π 2χL 4π R02 ) = 4π R02 ρ L CL ΔT ( RDL ω ω
[4.250]
RDL is the diffusion length already defined by relation [4.228]. The ratio of the heat given to the bubble to increase the temperature on that leaving the bubble by thermal conduction is: rQ =
⎛ ρ C ⎞⎛ R ⎞ Q = 3π ⎜ L L ⎟ ⎜ DL ⎟ ΔQG ⎝ ρG CG ⎠ ⎝ R0 ⎠
[4.251]
In the range rQ >> 1 , the thermodynamic process is practically isothermal since heat conduction is so fast that any temperature change in the bubble is efficiently inhibited. Conversely, in the range, rQ << 1 , the process is adiabatic since heat transfer has no time to take place. This characteristic ratio is plotted in Figure 4.60 for air bubbles in water. In a very broad domain of frequencies and bubble radii, the process is largely isothermal.
Plane acoustical waves in pipe systems
345
Figure 4.60. characteristic ratio of heat transfer versus R0 / RDL
4.4.3.2 Dispersive model accounting for the bubble vibrations As the frequency range of interest is increased, the quasi-static assumption used in the last subsection is no longer valid and the dynamics of the gas bubbles must be accounted for. The small vibration of a bubble according to the breathing mode is governed by the linearized version of equation [2.84], first written as: ρ L 0 R0
∂ 2δ R − δ pG = −δ pL ∂ t2
where the subscript
(0)
[4.252]
is used to indicate that the value of the related quantity is
taken at the static state of equilibrium and the symbol δ is used here to designate a real (not virtual) variation of small amplitude about it. The linearized variation of gas pressure is related to that of the bubble radius through the isothermal compressibility law, which gives: δ pG = −
3P0 δR R0
[4.253]
Hence, the bubble vibration is governed by the following equation: ⎛ ∂ 2δ R ⎞ ρ L 0 R0 ⎜ + ω 02δ R ⎟ = −δ pL 2 ⎝ ∂t ⎠
[4.254]
346
Fluid-structure interaction
where the natural pulsation of the isothermal breathing mode is: ω0 =
1 R0
3P0 ρL0
[4.255]
On the other hand, the liquid is governed by the mass equation. As outlined in [CRI 92], remarkably simple and enlightening results are obtained, if one assumes that the bubbles are all of the same size and if a few minor simplifications are made concerning the homogenized fluid density and the quasi-static homogeneous sound velocity. Neglecting the contribution of the gas to the homogenized fluid density, relation [4.241] reduces to: ρ H ρ L (1 − α H )
[4.256]
Therefore a small variation of the homogeneous density about the state of static equilibrium is: δρ H = (1 − α H 0 ) δρ L − ρ L 0δα H
[4.257]
The linear equation of continuity for the homogeneous fluid is: ∂ρ H ∂v + ρ H0 =0 ∂t ∂x
[4.258]
where in the plane wave motion, the liquid and the gas, hence the mixture also, are assumed to oscillate at the same longitudinal velocity v. With the aid of [4.256] and [4.257], equation [4.258] is rewritten as:
(1 − α H 0 )
∂δρ L ∂v ∂δα H + (1 − α H 0 ) ρ L0 − ρ L0 =0 ∂t ∂x ∂t
[4.259]
Denoting n the number of gas bubbles per unit volume, the void fraction is simply: αH 0 =
4π R03n0 4π R03δ n α δ n 3α δ R ⇒ δα H = + 4π R02 n0δ R = H 0 + H 0 3 3 n0 R0
[4.260]
It follows immediately that: ∂δα H 3α H 0 ∂δ R α H 0 ∂δ n = + ∂t R0 ∂ t n0 ∂ t
[4.261]
Assuming that the number of bubbles is not changed (incondensable gas), the following linear continuity equation is obtained: ∂n ∂v + n0 =0 ∂t ∂x
[4.262]
Plane acoustical waves in pipe systems
347
Substituting relations [4.261] and [4.262] into the continuity equation for fluid density [4.241], the latter is transformed into:
(1 − α H 0 )
∂δρ L 3α H 0 ρ L0 ∂δ R ∂v − + ρ L0 =0 ∂t R0 ∂t ∂x
[4.263]
On the other hand, the linear momentum equation for the liquid is:
(1 − α H 0 ) ρ L
0
∂v ∂pL + =0 ∂t ∂ x
[4.264]
As in the case of a single phase fluid, particle velocity is easily eliminated between the continuity and the momentum equations. Then, using the linear compressibility law for the liquid, the fluctuating liquid density is also eliminated. The final result is the following wave equation: ∂ 2 pL ⎛ 1 − α H 0 ⎞ ∂ 2 pL 3α H 0 (1 − α H 0 ) ρ L0 ∂ 2 R −⎜ + =0 ⎟ ∂ x 2 ⎝ cL ⎠ ∂ t 2 ∂ t2 R0 2
[4.265]
where the derivatives of the variations are identified with the corresponding quantities themselves. In the general case, the time and space derivatives of the liquid pressure can be expressed in terms of the time and space derivatives of the bubble radius through equation [4.254]. The result is written as: 2
2 2 3α H 0 (1 − α H 0 ) ∂ 2 R ⎛ 1 − αH 0 ⎞ ⎛ ∂4 R ⎛ ∂4R 2 ∂ R⎞ 2 ∂ R⎞ + =0 ⎜ ⎟ ⎜ 4 + ω0 2 ⎟ − ⎜ 2 2 + ω0 ⎟ ∂ t ⎠ ⎝ ∂ t ∂x ∂ x2 ⎠ ∂ t2 R02 ⎝ cL ⎠ ⎝ ∂ t
[4.266]
Assuming a harmonic travelling wave solution δ R exp i (ω t − kx ) , the following dispersion equation is finally obtained, which gives directly the phase velocity of the waves as a function of frequency: 2
1 k 2 ⎛ 1 − αH 0 ⎞ 1 = 2 =⎜ ⎟ + 2 2 2 cψ ω ⎞⎛ ω2 ⎞ R0 ω 0 ⎝ cL ⎠ ⎛ ⎜ ⎟ ⎜1 − 2 ⎟ ⎝ 3α H 0 (1 − α H 0 ) ⎠ ⎝ ω 0 ⎠
[4.267]
Relation [4.267] can be further simplified by using the value of ω 0 given in [4.255] and the isothermal value of the speed of sound in the gas. Then: R02ω 02 P0 ρG 0 cG2 = cH2 = 3α H 0 (1 − α H 0 ) ρ L 0α H 0 (1 − α H 0 ) ρ L 0α H 0 (1 − α H 0 )
[4.268]
The last simplification stems from the following simplified form of the homogeneous speed of sound [4.244]:
348
Fluid-structure interaction 1/ 2
⎛ ⎞ ρG 0 cH cG ⎜⎜ ⎟⎟ ⎝ ρ L 0α H 0 (1 − α H 0 ) ⎠
[4.269]
Then, the phase velocity of the bubbly fluid is finally expressed as: ⎛ ⎞ 2 ⎜ ⎟ ⎛ 1 − αH 0 ⎞ ω 1 ⎟ + cψ = = ⎜ ⎜ ⎟ k ⎜ ⎝ cL ⎠ ω2 ⎞ ⎟ 2 ⎛ cH ⎜ 1 − 2 ⎟ ⎟⎟ ⎜⎜ ⎝ ω0 ⎠ ⎠ ⎝
−1/ 2
[4.270]
As shown in Figure 4.61, sound propagation in the bubbly fluid is dispersive, except if the frequency of the wave is much less, or on the contrary, much higher than the natural frequency of the breathing mode of the bubbles. In the low frequency range the sound velocity tends to the quasi-static value cH as appropriate. As already stated in the last subsection, only the compressibility of the gas is of importance in the low frequency range. In terms of bubble vibration, this means that so long as ω remains sufficiently smaller than ω 0 , the response of the bubble to a pressure change lies in the quasi-static range. Accordingly, the inertia term in the bubble equation [4.254] can be disregarded, and, as an immediate corollary, it is found that δ pL = δ pG . Substituting the value [4.253] of the pressure change into the wave equation [4.265] yields: 2 3P0 ∂ 2 R ⎛ 3P0 ⎛ 1 − α H 0 ⎞ 3α H 0 (1 − α H 0 ) ρ L 0 ⎞ ∂ 2 R ⎜ ⎟ 2 =0 − ⎜ ⎟ + ⎟ ∂t R0 ∂ x 2 ⎜ R0 ⎝ cL ⎠ R0 ⎝ ⎠
[4.271]
after a few elementary algebraic steps, the equation [4.271] reduces to the non dispersive wave equation: ∂2 R 1 ∂2 R − =0 ∂ x 2 cH2 ∂ t 2
[4.272]
Then, as frequency is increased, the fluid inertia associated with the breathing mode of vibration of the bubbles becomes increasingly important. As no additional stiffness is delivered to the system, the phase speed of sound decreases and even becomes imaginary within a fairly large range of frequencies extending from ω 0 up to ω 0 cL / cH . As a consequence, no sound waves can propagate within that band of frequencies. Above ω 0 cL / cH , propagation is again possible. Near the threshold, phase speed is much larger than the corresponding value in pure liquid, then cψ decreases steadily, tending asymptotically to the remarkable value: cψ
lim ω →∞
=
cL 1 − αH 0
[4.273]
Plane acoustical waves in pipe systems
349
Figure 4.61. Real part of the phase velocity versus frequency
The asymptotic result [4.273] means that as frequency is sufficiently increased the amplitude of the breathing vibration vanishes. As a consequence, due to the wave motion, the bubbles are oscillating at the longitudinal velocity v in a passive way, that is affording negligible compressibility and inertia. Therefore, the only effect of bubbles is to diminish the effective density and increase the stiffness of the medium through the void fraction effect. With the aid of the relation E L / ρ L 0 = cL2 , the asymptotic result [4.273] is immediately recovered: cψ =
4.4.4
EL
(1 − α H 0 )
2
ρL0
[4.274]
Speed of sound of a fluid contained within elastic walls
The gas-liquid mixture studied in the last subsection may be viewed as a compound compressible medium comprising two distinct phases intimately mixed together and highly contrasted with respect to their density and elastic properties. Other examples of such compound materials, involving this time a fluid and a solid are also of large interest in practice. In porous materials the solid and fluid phases are also intimately mixed together, favouring thus friction and interfacial exchanges of heat and mechanical energy. As porous materials of various kinds are commonly used to dissipate sound energy into heat, the subject is addressed in many textbooks, see in particular [ALL 93]. Here, we are interested in a much simpler compound
350
Fluid-structure interaction
material, which is defined as a fluid enclosed by solid elastic walls. As in the case of the gas-liquid mixture, the elastic dilatation of the solid envelope induced by the pressure waves tends to increase the equivalent compressibility of the solid-liquid compound and correlatively to decrease the speed of sound with respect to the intrinsic value for the fluid alone (infinite extent of fluid, or fluid contained by rigid walls). Contrasting with the gas-liquid mixture, the volume change that matters is that of the less compressible phase. So, at first sight at least, it could be thought that the effect is very small, based on the very large value of the ratio of Young’s modulus of stiff materials like steel, versus gases and even liquids. Actually, this is not necessarily the case because, in the present problem, the solid phase is shaped as a thin plate or shell structure. Therefore, dilatation of the solid depends not only on the Young’s modulus of the material but also on the structural properties of the solid body. The problem is worked out just below for the example of a fluid contained within a thin circular cylindrical shell, see Figure 4.62. This structural element is particularly convenient for mathematical simplicity and relevant for application in many piping systems.
Figure 4.62. Circular cylindrical tube with elastic walls
Let M o designate the mass of fluid contained per unit length of pipe in the absence of any acoustical wave. Starting from this state of reference, due to the loading by sound pressure δ P , M o is changed into: M = ( ρ 0 + δ ρ ) S0 + ρ 0δ S = M 0 + δ ρ S0 + ρ 0δ S
[4.275]
The first variation is induced by the fluid compressibility and the second by the dilatation of the walls. Accordingly, the compound system behaves as an equivalent fluid, density of which is varied by the following amount: δ ρ = δ ρ + ρ 0δ S / S0
[4.276]
As a consequence, the equivalent speed of sound of the fluid, denoted c, is related to the value co of the fluid alone by the following relationships:
Plane acoustical waves in pipe systems
c2 =
δp δp and c02 = ⇒ δρ δ ρ0
⎡ ρ δS⎤ c / c0 = ⎢1 + 0 ⎥ δρ 0 S0 ⎦ ⎣
−1/ 2
⎡ ρ c2 δ S ⎤ = ⎢1 + 0 0 δ p S0 ⎥⎦ ⎣
−1/ 2
351
[4.277]
For quantifying the structural effect on the speed of sound of the fluid, it is necessary to relate the relative change in cross-sectional area of the tube to the acoustical pressure, which is easily worked out in the case of an elastic circular cylindrical shell. Though the sound pressure field is a dynamic load, so long as the frequency of the sound waves is less than the first breathing mode of vibration of the shell, the structural response remains in the quasi-static range. Furthermore, provided δ p is not too large, the calculation can be carried out by assuming small elastic strains. It follows that: δ S / So = 2δ R / Ro = 2ε
[4.278]
Figure 4.63. Static equilibrium of a wall element subjected to a uniform inner pressure field
Here ε << 1 stands for the membrane strain of the thin shell. The problem can be treated by using the Love’s shell equations, see [AXI 05] Chapter 8. However, provided the acoustic wavelengths are much larger than the shell radius, only the radial equation is needed, which can be easily written down directly. As shown in Figure 4.63, an infinitesimal element of wall is subjected to the pressure force Fp acting in the radial direction n (θ ) , and the elastic hoop stresses Nθθ acting in the tangential direction t (θ ) . These forces are expressed as: 1 Fp = δ pRo dθ n (θ ) ; N θθ = E s ε ht (θ ) 2
[4.279]
where h is the wall thickness, and Es is the Young’s modulus of the shell material. The force balance in the radial and tangential directions immediately follows as:
352
Fluid-structure interaction
Fp = 2N θθ sin(dθ / 2) = Nθθ dθ Nθθ ( cos(dθ / 2) − cos(dθ / 2) ) = 0
[4.280]
Substituting the elastic stress law [4.279] into the radial equation [4.280] allows one to relate the strain to the pressure: ε=
R0 δp Es h
[4.281]
Substituting the elastic strain law [4.281] into [4.278] and [4.277] successively, one obtains: c ⎡ 2 Ro E f ⎤ = 1+ c0 ⎢⎣ h Es ⎥⎦
−1/ 2
[4.282]
The final result [4.282] brings out the two important parameters which control the relative increase of the compound fluid compressibility, namely the ratio of the Young’s modulus of the fluid on that of the solid, and the reduced thickness of the shell. The lowering of the speed of sound can be significant, provided E f is not too small in comparison with Es and provided h is much smaller than Ro . As an illustrative example, let us select the following structural and fluid properties, which correspond to a thin steel shell filled with water: Es = 2.1011 Nm -2 ; h =2 mm; Ro = 20 cm; ρ0 = 103 kgm -3 ; c0 = 1 500 ms-1 .
The speed of sound within the pipe is found to be c 832 ms −1 instead of 1 500 ms-1 . The lowering is even much more pronounced in the case of water-like liquids contained within biomaterials, blood within arteries and veins for instance.
Chapter 5
3D Sound waves
In this chapter, the concepts described in the context of plane sound waves are extended to two or three-dimensional cases, including harmonic and transient sound waves. In contrast with the usual order of presentation in the general context of the present book, it is found appropriate to describe first the acoustic modes in 3D enclosures provided with perfectly reflective boundary conditions and then guided waves, before addressing the subject of acoustic sources and sound radiation in an infinite medium. As a preliminary, a few basic aspects of sound wave reflection and refraction at the interface between two elastic media are presented, which serve to clarify the physical feasibility of the idealized boundary conditions used for the theoretical analysis of the standing waves. Then we proceed by describing the properties of acoustic modes successively in rectangular, cylindrical and finally in spherical enclosures, which are the three basic geometries which allow one to solve the modal problem analytically by using the separation variables method and to get familiarized with the special functions suited to handling problems in cylindrical and then in spherical coordinates. The object of the second section is to describe sound waves which are guided along a given direction by perfectly reflecting boundaries, such as lateral fixed walls. The concept of guided mode waves is introduced first as a natural extension of acoustic modes, by letting one dimension unbounded, which defines the axis of the guidewave. Guided mode waves can be understood as resulting from the interference pattern of pairs of plane waves travelling at oblique incidence with respect to the axis of the waveguide. The last viewpoint is well suited to clarify the physical meaning of the dispersive nature of guided waves. The third section is concerned with the problem of modelling acoustic sources and computing acoustic response either in a finite or infinite medium. Beside the modal projection methods, Green’s functions and related
354
Fluid-structure interaction
methods based on the Kirchhoff-Hemholtz integral theorem are especially attractive for handling large enclosure or even open space problems. On the other hand, as is well known, sound waves are often excited by the motion of a solid. In the present chapter the motion of the solid is assumed to be prescribed independently of the acoustical response of the fluid medium. In practice, this corresponds typically to a sufficiently compressible and light fluid, such as air at STP, driven by a sufficiently stiff and heavy solid. 5.1. 3D Standing sound waves (acoustic modes) 5.1.1
Modal equations and general properties of acoustic modes
5.1.1.1 Interface separating two media and boundary conditions Before entering into the subject of modal analysis in acoustics, it is of interest to extend first the reflection and transmission laws of plane waves at normal incidence to the case of oblique incidence. The main object is to clarify the conditions which hold at the boundaries of the fluid volume, focusing on those which can be idealized as purely reflective, hence conservative in nature.
Figure 5.1. Reflected and transmitted plane wave at a plane interface between two fluids
3D Sound waves
355
As shown in Figure 5.1, the z = 0 plane of a Cartesian frame is assumed to be the interface between two fluids of distinct specific impedances. A sound plane wave propagates along the direction specified by the unit incident vector: i = sin θi i + cos θi k [5.1] where the angle of incidence θi is that angle made by the direction of the incident plane wave with the unit vector k normal to the plane interface. The incident wave is assumed to be partially reflected back into the first medium along the r direction of reflection, defined as: r = sin θ r i − cos θ r k [5.2] and partially transmitted to the second medium along the tr direction of refraction, defined as: tr = sin θ tr i + cos θ tr k [5.3] To establish the reflection and refraction laws of elastic waves, it suffices to deal with harmonic waves. Extension to more complicated time variations follows without additional difficulty by using the Laplace, or the Fourier transform. Furthermore, the problem is much simpler in the case of fluids than that of solids ([cf. AXI 05]), since no shear elastic waves can occur in a fluid, as already emphasized in Chapter 1. So, the incident, reflected and transmitted pressure waves are written as: ⎛ ( x sin θi + z cos θi ) ⎞ + pi = pi( ) exp iω ⎜ t − ⎟ c1 ⎝ ⎠ ⎛ ( x sin θ r − z cos θ r ) ⎞ − pr = pr( ) exp iω ⎜ t − ⎟ c1 ⎝ ⎠
[5.4]
⎛ ( x sin θtr + z cos θ tr ) ⎞ + ptr = ptr( ) exp iω ⎜ t − ⎟ c2 ⎝ ⎠
The condition of force equilibrium at the mass-less interface (z = 0) yields the pressure balance: pi( ) e − iω x sinθi / c1 + pr( ) e − iω x sinθ r / c1 = ptr( ) e − iω x sinθtr / c2 +
−
+
∀x
[5.5]
Since the balance [5.5] must hold independently from the value of the phase angles, the following reflection and transmission laws are necessarily verified: sin θi sin θ r sin θtr = = ⇔ θi = θ r c1 c1 c2
⎛c ⎞ ; θtr = sin −1 ⎜ 2 sin θi ⎟ c ⎝ 1 ⎠
[5.6]
356
Fluid-structure interaction
The angular relations [5.6] are thus found to be the same as in optics and, as a corollary, total reflection occurs at incidence beyond the critical value: θc = sin −1 ( c1 / c2 )
[5.7]
Using the refraction law [5.6] it is immediately concluded that θ r cannot be a real angle in the range θi > θ c . Actually the condition θi > θ c implies sin θ tr > 1 and the cosine is purely imaginary: cos θ tr = ± 1 − sin 2 θ tr = ±iγ
[5.8]
Substituting [5.8] into the transmitted wave [5.4], yields: (+)
ptr = ptr e
⎛ ( x sin θ tr ± izγ ) ⎞ iω ⎜⎜ t − ⎟⎟ c2 ⎝ ⎠
(+)
= ptr e
±
⎛ x sin θ tr ⎞ ωγ z iω ⎜ t − ⎟ c2 ⎠ c2 ⎝
e
[5.9]
The only physically meaningful solution corresponds to the minus sign which represent an evanescent wave travelling at speed c2 in the Ox direction and whose amplitude decreases along the Oz direction exponentially with the scale length λ2γ / 2π where λ2 is the wavelength in the second medium.
Figure 5.2. Equivalent tubes for the reflected and transmitted plane waves
The relation between the refection and transmission coefficients of the pressure waves, already established as 1+R = T in the case of normal incidence (see equation [4.20]) is found here to hold at any angle of incidence. However, there is a difference between the oblique versus the normal incidence, because, in an inviscid fluid, only the normal component of particle velocity must be continuous across the
3D Sound waves
357
interface. Any discontinuity in a tangential direction is of no consequence so long as viscous forces are discarded. The condition of continuity of the normal velocities at the interface is written as: ui( ) cos θi + ur( ) cos θ r = utr( ) cos θ tr +
−
+
[5.10]
Using the reflection law θi = θ r and the relations [4.17] defining the specific impedances for travelling waves, it is not difficult to extend the relations [4.21] and [4.22] to the case of oblique incidence. The result is: 1− R = R=
Z2(
Z2(
Z1(
sp )
( sp )
Z2 sp ) sp )
cos θtr cos θi
T
cos θi − Z1(
cos θi + Z1(
sp ) sp )
cos θ tr cos θ tr
; T=
Z2(
sp )
2Z2(
sp )
[5.11]
cos θi
cos θi + Z1(
sp )
cos θ tr
Physical interpretation of such results can be clarified by referring to the case of plane waves in piping systems. As indicated in Figure 5.2, nothing prevent us from drawing mentally a system of three tubes in which the incident, reflected and refracted waves propagate. The tubes are connected to each other at the interface between the two media. Denoting S0 the area of the oblique cross-section common to each tube, the areas of the normal cross-sections of the incident, reflected and transmitted tubes are found to be: Si = S0 cos θi = Sr
; Str = S0 cos θ tr
[5.12]
Therefore, in line with the concept of pipe impedance introduced in Chapter 4 by the relations [4.16], for the present problem we define the following impedances: Z1 =
Z1( ) cos θi sp
; Z2 =
Z2( ) cos θ tr sp
[5.13]
where S0 is tacitely assumed to be a unit area. Clearly, if the specific impedances are replaced by the tube impedances [5.13], the reflection and transmission factors can be expressed independently of the angle of incidence as: R=
Z2 − Z1 Z2 + Z1
; T=
2Z2 Z2 + Z1
[5.14]
The asymptotic cases of most interest to us concern the conservative boundary conditions of total wave reflection. In this respect, it can be noticed that, except for a few specific values of the angle of incidence, namely the grazing incidence θi = 90° and the angle of intromission θ 0 such that Z2(
sp )
cos θ 0 = Z1(
sp )
cos θ tr , the same
358
Fluid-structure interaction
conclusions as in the case of normal incidence hold. Namely, if Z2 << Z1 , relations [5.14] tend asymptotically to R = −1 and T = 0 ; thus the corresponding boundary condition is a pressure node, also called pressure release condition in many textbooks on acoustics. Such a behaviour holds at grazing incidence, whatever the value of the specific impedance of the second medium may be, because as θ i tends to 90°, Z1 tends to infinity. Even if incidence is below the critical angle for total reflection, acoustic power transmitted to the second medium is negligible. In the opposite case, Z1 << Z2 , relations [5.14] tend asymptotically to R = +1 and T = 2 , that is a node of normal velocity, which is also called fixed wall condition. To conclude this subsection it is worth mentioning that at the intromission angle θ 0 , if it exists, perfect transmission of sound through the interface occurs. In what follows only the asymptotical cases leading to total wave reflection with negligible energy transmission are considered. The case of sound transmitting boundaries is postponed to Chapter 7. 5.1.1.2 Wave equation expressed in terms of displacement field The local equations which govern the linear and conservative sound waves in a three dimensional quiescent fluid were already established in Chapter 1, subsection 1.3.1 based on the linearization of the Euler and the compressibility equations resulting in the system [1.55], repeated here for convenience as: p = ρ c 2f
∂X f ∂ρ + ρ f div = 0 ⇔ ρ + ρ f div X f = 0 ∂t ∂t 2 ∂ Xf grad ρf p=0 + ∂t2
[5.15]
where no external load is assumed to be present within the fluid volume. Acoustical sources will be considered later, in section 5.3. As also demonstrated in Chapter 1, elimination of the pressure between the mass and the momentum equations results in the wave equation [1.56]. It is recalled, if necessary that the subscript ( 0 ) used in Chapter 1 to mark the quiescent state of reference of the fluid has been then changed into ( f ) , starting from Chapter 2, to contrast the solid and fluid quantities. Here, we consider a finite volume (Vf
)
of fluid limited by a closed surface (S f ) . The modal
problem can be written as: −grad ρ f c 2f div X f − ω 2 ρ f X f = 0
(
+ ( C.B.C. )
)
[5.16]
3D Sound waves
359
where (C.B.C.) denotes an appropriate set of conservative boundary conditions, which take the form of a conservative impedance verified at any point of (S f ) . Here, it is expressed in the same way as in solid mechanics, that is as a linear and homogeneous relationship between the normal components of the stress and the displacement variables: α div X f n + β X f .n =0 [5.17]
(
)
(S f )
where n denotes the unit vector, normal to (S f ) , pointing from the fluid to the
outside. As particular cases of special importance, depending on whether α, or β is equal to zero, the boundary is fixed or free, respectively. As indicated in subsection 5.1.1.1, a fixed boundary can be achieved in practice by limiting the fluid by a fixed and sufficiently thick wall, whereas a free boundary condition is practically achieved if a liquid is limited either by a free level (cf. Chapter 3), or by a thin and very flexible wall, water contained in a thin glass or plastic container, for instance (cf. Chapter 4). On the other hand, an elastic impedance characterized by the stiffness coefficient per unit area κ S corresponds to α = 1 and β = −κ S , whereas an inertial impedance of mass coefficient per unit area μS corresponds to α = 1 and β = ω 2 μS .
As expected, the system [5.16] is of the same canonical form as in the case of an elastic solid and is thus written here as: ⎡⎣ K f − ω 2 M f ⎤⎦ ⎡⎣ X f ⎤⎦ = 0 [5.18] The mass operator of the fluid is: Mf
[ ]= ρf
and the stiffness operator is: K f [ ] = −grad ( ρ f c 2f div [ ])
[5.19]
[5.20]
Furthermore, as the fluid oscillates about a static and stable, or neutral, state of equilibrium, these operators must be self-adjoint and positive (cf. [AXI 05], Chapter 3). Therefore, the solutions of the modal problem [5.16] are endowed with the same properties as in the case of an elastic solid. Due to the three dimensional geometry of the fluid volume, the modal quantities are written as: X ϕ (,m),n ( r ) ; {ω ,m ,n } ; {K ,m ,n } ; {M ,m ,n } ; , m, n = 1, 2,3,.. [5.21]
{
}
360
Fluid-structure interaction
X Where the superscripted (X) marks the fact that ϕ (,m),n is a normalized displacement
field, in contrast with ϕ (,m),n which is normalized pressure field. Each one of the p
integer indices , m, n characterizes a direction in the 3D-Euclidean space, which is orthogonal to the two others. The natural pulsations ω ,m ,n are positive, or zero if the fluid can move freely without expanding or contracting. The mode shapes ϕ ,m ,n ( r ) are real and comply with the following orthogonality conditions: ≥ 0 if 1 = 2 , m1 = m2 , n1 = n2 ⎧K X X ϕ (1 ,m)1 ,n1 ( r ) , K f ⎣⎡ϕ (2 ,)m2 ,n2 ( r ) ⎦⎤ = ⎨ ,m ,n 0 otherwise (Vf ) ⎩ > 0 if 1 = 2 , m1 = m2 , n1 = n2 ⎧M X X = ⎨ ,m ,n ϕ (1 ,m)1 ,n1 ( r ) , M f ⎣⎡ϕ (2 ,m) 2 ,n2 ( r ) ⎦⎤ 0 otherwise (Vf ) ⎩
[5.22]
It is recalled that the functional scalar product is defined as: X X ϕ (1 ,m)1 ,n1 ( r ) , ϕ (2 ,m) 2 ,n2 ( r )
⌠
(Vf )
= ⎮⎮
⌡(Vf
)
X X ϕ (1 ,m)1 ,n1 ( r ) .ϕ (2 ,)m2 ,n2 ( r ) dV
[5.23]
the dot appearing in the integral denoting the scalar product in the 3D-Euclidean space used to describe the fluid volume and the fluid Eulerian fields. Validity of the relations [5.22] can be verified as outlined below. X X Let ϕ (1 ,m) 1 ,n1 ( r ) and ϕ (2 ,)m2 ,n2 ( r ) be two mode shapes associated with the natural angular frequencies ω1 ,m1 ,n1 and ω 2 ,m2 ,n2 respectively. As a consequence, the following weighted integral relation must hold:
( (
)
⌠ ⎮ ⎮ ⎮ ⌡ Vf
X X X ϕ (1 ,m) 1 ,n1 . grad ρ f c 2f div ϕ ( 2 ,)m2 ,n2 + ω22 ,m2 ,n2 ρ f ϕ ( 2 ,)m2 ,n2 dV =
⌠ ⎮ ⎮ ⎮ ⎮ ⌡ Vf
X X X ϕ ( 2 ,)m2 ,n2 . grad ρ f c 2f div ϕ (1 ,m) 1 ,n1 + ω21 ,m1 ,n1 ρ f ϕ (1 ,m) 1 ,n1 dV
( )
( )
( (
)
)
)
[5.24]
Using the elastic law (see first line of the system [5.15]), the condition of symmetry [5.24] may be conveniently transformed into: ⌠ ⎮ ⎮ ⌡(Vf
(ω
)
X X ϕ (1 ,m)1 ,n1 .grad p 2 ,m2 ,n2 − ϕ (2 ,m) 2 ,n2 .grad p1 ,m1 ,n1 dV =
2 2 , m2 , n2
−ω
2 1 , m1 , n1
)
⌠ X X ⎮ ρ 0ϕ (1 ,m)1 ,n1 .ϕ (2 ,)m2 ,n2 ⎮ ⌡(Vf )
[5.25] dV
3D Sound waves
361
The left-hand side of [5.25] is further transformed by using the vectorial identity: X .grada = div aX − adivX [5.26]
( )
It gives: ⌠ ⎮ ⎮ ⎮ ⌡(Vf ⌠ ⎮ ⎮ ⎮ ⌡(Vf
(ω
)
)
X X div ⎡⎣ p 2 ,m2 ,n2 ϕ (1 ,m) 1 ,n1 − p1 ,m1 ,n1ϕ (2 ,)m2 ,n2 ⎤⎦ dV − ⎡ p ,m ,n div ϕ ( X,m) ,n − p ,m ,n div ϕ ( X,)m ,n ⎤ dV = 1 1 1 1 1 1 2 2 2 ⎦ ⎣ 2 2 2
2 2 , m2 , n2
⌠
− ω21 ,m1 ,n1 ) ⎮⎮
⌡(Vf
)
[5.27]
X ρ f ϕ (1 ,m) 1 ,n1 .ϕ 2 ,m2 ,n2 dV
The integral in the first line is readily transformed into a surface integral which vanishes identically because of the boundary conditions. For instance, if (S f ) is fixed, the normal displacement vanishes on (S f ) . If (S f ) is free, the pressure is zero on (S f ) . Extension of the proof to any elastic or inertial boundary condition
proceeds as already described in [AXI 05] and is left to the reader as an exercise. The integral in the second line vanishes identically since, by using the first and second equations [5.15], it can be written in terms of pressure solely as: ⌠ ⎮ ⎮ ⎮ ⎮ ⌡(Vf
)
1 ⎡ − p 2 ,m2 ,n2 p1 ,m1 ,n1 + p1 ,m1 ,n1 p 2 ,m2 ,n2 ⎤⎦ dV ≡ 0 ρ f c 2f ⎣
[5.28]
Therefore, if ω 1 ,m1 ,n1 differs from ω 2 ,m2 ,n2 the integral in the third line of [5.27] necessarily vanishes and the conditions [5.22] hold. However, as in the case of solids, distinct acoustical modes can vibrate at the same natural frequency. If such is the case, to a n-multiple eigenvalue of the modal equation corresponds a vectorial subspace of dimension n of eigenvectors (i.e. the mode shapes) which is orthogonal to the complementary subspace, as already described in [AXI 04] in the context of discrete systems. Furthermore, within the n-subspace one can always select n eigenvectors orthogonal to each other, hence linearly independent. As a consequence of paramount importance both in theory and practice, the mode shapes can always be used to determine a vector basis, orthonormal with respect to the stiffness and mass operators, in which the solution X f ( r ; t ) of any forced problem complying with the same homogeneous boundary conditions as the modal problem can be expanded as the modal series:
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∞ X f (r,t ) = ∑ =1
∞
∞
∑ ∑q m =1
n =1
,m , n
( t ) ϕ (,Xm),n ( r )
[5.29]
The time functions q,m ,n ( t ) , termed modal displacements, are the components of the displacement field X f ( r ; t ) in the modal coordinate system. To conclude this subsection, it is noticed that if the fluid is homogeneous, the modal equation [5.16] simplifies into: ω2 ρ f c 2f ΔX f + ρ f ω 2 X f = 0 ⇔ ΔX f + 2 X f = 0 cf
[5.30]
Since the mass operator of the modal problems governed by equation [5.30] is constant, the orthogonality relations [5.22] simplify into the direct orthogonality between the modes shapes: X X X X = ρ f ϕ (1 ,m)1 ,n1 ( r ) , ϕ (2 ,)m2 ,n2 ( r ) ⇒ ϕ (1 ,m) 1 ,n1 ( r ) , ρ f ⎡⎣ϕ (2 ,)m2 ,n2 ( r )⎤⎦ (Vf ) (Vf ) X X = 0 unless 1 = 2 , m1 = m2 , n1 = n2 ϕ (1 ,m) 1 ,n1 ( r ) , ϕ (2 ,)m2 ,n2 ( r )
[5.31]
(Vf )
5.1.1.3 Wave equation expressed in terms of pressure As shown in Chapter 1, subsection 1.3.1.3, provided the fluid is homogeneous, the wave equation can be expressed in terms of pressure solely and the modal problem can be formulated as: Δp +
ω2 p=0 c 2f
[5.32]
+ ( C.B.C )
It may be noted that equation [5.32] is very similar to equation [5.30], except that the field variable now is a scalar instead of a three dimensional vector. Another difference is that when shifting from the displacement to the pressure field variable, the physical meaning of the operators is exchanged. Indeed, the Laplacian operating on X f is interpreted as a stiffness operator, while in [5.32] it describes the fluid inertia as amply demonstrated already in Chapters 2 and 3. On the other hand, the second operator ω 2 / c 2f = k 2 , which accounts for fluid inertia in equation [5.30], describes fluid elasticity in equation [5.32] as indirectly shown in Chapter 1, subsection 1.3.3.4 through the physical meaning given to the compressibility parameter. Actually, if the frequency tends to zero, or in an equivalent way, the wavelength tends to infinity, the second term of equation [5.32] can be dropped and correlatively the fluid elasticity discarded. All these properties are implicitely
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363
contained in the statement that there is a relationship of duality between the formulations in terms of either displacement or pressure field. On the other hand, the boundary conditions can be written as: ap + bn.grad p = 0 [5.33] Depending on whether a, or b is equal to zero, the boundary is fixed or free, respectively. The pressure mode shapes are denoted:
{ϕ ( )
p ,m ,n
( r )}
, m, n = 1, 2,3,..
[5.34]
X p Once more, it is stressed that ϕ ( ) is a scalar while ϕ ( ) is a vector quantity. On the other hand, since the fluid is supposed homogeneous, the relations of orthogonality [5.22] are replaced by the relation of direct orthogonality:
p ϕ 1 ,m1 ,n1 ( r ) , ϕ ( 2 ,)m2 ,n2 ( r ) ( p)
⎧⎪ ϕ = ⎨ , m ,n (Vf ) ⎪ ⎩
2
> 0 if 1 , m1 , n1 = 2 , m2 , n2 = , m, n 0 otherwise
[5.35]
Validity of the relations [5.35] can be checked in the same way as was done for the relations [5.22]. However, it is found interesting to derive it in a slightly distinct way as a consequence of the self adjointness of the modal problem [5.32]. Applying Green’s identity (for the proof see Appendix A3) to two modes shapes, which by definition comply with the boundary conditions [5.33], we obtain: ⌠ ⎮ ⎮ ⎮ ⌡(Vf ⌠ ⎮ ⎮ ⎮ ⌡(S f
)
)
(ϕ ( )
p 1 ,m1 ,n1
)
Δϕ (2 ,)m2 ,n2 − ϕ ( 2 ,)m2 ,n2 Δϕ (1 ,m) 1 ,n1 dV = p
p
p
p p p ϕ 1 ,m1 ,n1 gradϕ (2 ,)m2 ,n2 − ϕ ( 2 ,)m2 ,n2 gradϕ (1 ,m) 1 ,n1 .nd S = 0
(
[5.36]
)
On the other hand, the following weighted integral relation must hold: ⌠ ⎮ ⎮ ⎮ ⌡(Vf ⌠ ⎮ ⎮ ⎮ ⌡(Vf
)
)
(ϕ ( )
p 1 ,m1 ,n1
)
Δϕ (2 ,)m2 ,n2 − ϕ (2 ,)m2 ,n2 Δϕ (1 ,m) 1 ,n1 dV + p
p
p
[5.37]
(ω
2 2 ,m2 ,n2
)
− ω21 ,m1 ,n1 ϕ (1 ,m) 1 ,n1ϕ ( 2 ,)m2 ,n2 dV = 0 p
p
At this step, reasoning to prove the orthogonality relation [5.35] follows the same lines as for the orthogonality relations [5.22].
364
5.1.2
Fluid-structure interaction
Analytical examples of acoustical modes
5.1.2.1 Rectangular enclosure
Figure 5.3. Rectangular enclosure filled with a homogeneous fluid
We consider an elastic and inviscid fluid filling a rectangular box, or enclosure. This particular geometry is chosen as a first analytical example, for convenience in using the Cartesian coordinate system specified in Figure 5.3. From the mathematical standpoint, the conservative boundary conditions to be fulfilled at the ‘walls’ of the box can be selected in various ways, in accordance with the conditions [5.17], or [5.33], depending on which field variable is selected to solve the modal problem. Here, it suffices to restrict the presentation to the case of a box limited by fixed and perfectly reflective walls. This kind of boundary condition can be realized physically to a good degree of approximation, provided the wall is sufficiently thick and made with a material selected as poorly absorbing and of specific impedance much higher than that of the fluid. This is typically the case of thick walled steel or stone boxes enclosing air.The modal problem written in terms of the displacement field X f is: ω2 ΔX f + 2 X f = 0 cf X f .i = X f .i =0 x =0 x = Lx Xf.j = Xf.j =0 y =0 y = Ly X f .k = X f .k =0 z =0
[5.38]
z = Lz
Its solution was already treated in the context of solid dilatational waves in [AXI 05] Chapter 1. Hence, a short presentation which emphasizes the most salient points will suffice here. First, the vector equation is transformed into a scalar eqation by using
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either a displacement potential Φ ( x, y , z ) , or a velocity potential Φ ( x, y , z ) , leading to: 2
⎛ω ⎜ ⎜ cf ⎝
⎞ ⎟ Φ + ΔΦ = 0 ⎟ ⎠
∂Φ ∂x
=
[5.39]
On the other hand, as X f = − gradΦ , the boundary conditions are expressed as:
x =0, Lx
∂Φ ∂y
= y =0, Ly
∂Φ ∂z
=0
[5.40]
z =0, Lz
The system [5.39], [5.40] is easily solved by separating the variables. The trial function Φ ( x, y , z ) = A(x)B(y)C(z) is substituted into [5.39] and one obtains: ⎛ω ⎜ ⎜ cf ⎝
2
⎞ A′′ B ′′ C ′′ + + =0 ⎟ + ⎟ A B C ⎠
[5.41]
Noting that A”/A can be a function of x only, B”/B of y only and C”/C of z only, it follows that all these quantities must be constants. Furthermore, owing to the geometry of the enclosure and the boundary conditions assumed, x,y,z are interchangeable, in contrast with the case of sloshing modes, cf. Chapter 3 subsection 3.4.2.1. So [5.41] leads to the dispersion equation: ⎛ω ⎜ ⎜ cf ⎝
2
⎞ 2 2 2 ⎟ − kx + k y + kz = 0 ⎟ ⎠
(
)
[5.42]
where the constants k x , k y , k z are the Cartesian components of a wave vector k .
Each one of the functions A(x) , B(y) , C(z) is found to be governed by the same differential equation as that which governs the plane waves in the same direction: d2A + k x2 A = 0; 2 dx
d 2B + k y2 B = 0; 2 dy
d 2C + k z2C = 0 2 dz
[5.43]
Whence the mode shapes expressed in terms of the displacement potential : ⎛ mπ y ⎞ ⎛ π x ⎞ ⎛ nπ z ⎞ Φ ,m ,n ( x, y , z ) = φ0 cos ⎜ ⎟ cos ⎜ ⎟ cos ⎜⎜ ⎟ , m, n = 0,1, 2,... ⎟ ⎝ Lz ⎠ ⎝ Lx ⎠ ⎝ Ly ⎠
where φ0 is an arbitrary constant used to specify the norm.
[5.44]
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Fluid-structure interaction
The corresponding natural pulsations are: 2
ω , m ,n = c f
2
⎛ π ⎞ ⎛ mπ ⎞ ⎛ nπ ⎞ ⎟ +⎜ ⎜ ⎟ +⎜ ⎟ ⎝ Lx ⎠ ⎜⎝ Ly ⎟⎠ ⎝ Lz ⎠
2
, m, n = 0,1, 2,...
[5.45]
Incidentally, it may be noted that the solution given by [5.44] and [5.45] is identical whether it is expressed in terms of pressure or in terms of potential. In terms of displacements, the mode shapes are written as: ⎧ ( x) ⎛ mπ y ⎞ ∂Φ ,m ,n ⎛ πφ0 ⎞ ⎛ π x ⎞ ⎛ nπ z ⎞ = ⎜− ⎪ ϕ ,m , n ( x , y , z ) = ⎟ cos ⎜ ⎟ sin ⎜ ⎟ cos ⎜⎜ ⎟ ⎟ ∂x ⎪ ⎝ Lz ⎠ ⎝ Lx ⎠ ⎝ Lx ⎠ ⎝ Ly ⎠ ⎪ ∂Φ ,m ,n ⎛ mπφ0 ⎞ ⎛ π x ⎞ ⎛ mπ y ⎞ ⎛ nπ z ⎞ ⎪ ( y) = ⎜− ⎟ cos ⎜ ⎟ cos ⎜ ⎨ϕ ,m,n ( x, y , z ) = ⎟ sin ⎜⎜ ⎟ ⎜ ⎟ ⎟ ∂y ⎝ Lz ⎠ ⎝ Lx ⎠ ⎝ Ly ⎠ ⎪ ⎝ Ly ⎠ ⎪ ⎪ ϕ ( z ) x, y , z = ∂Φ ,m ,n = ⎛ − nπφ0 ⎞ cos ⎛ π x ⎞ cos ⎛ mπ y ⎞ sin ⎛ nπ z ⎞ ⎜ ⎟ ⎜ ) ⎜ ⎟ ⎟ ⎜ ⎟ ⎪ ,m , n ( ⎜ ⎟ ∂z ⎝ Lz ⎠ ⎝ Lx ⎠ ⎝ Ly ⎠ ⎝ Lz ⎠ ⎩⎪
[5. 46]
and in terms of pressure as: ⎛ mπ y ⎞ ⎛ π x ⎞ ⎛ nπ z ⎞ p ϕ (,m),n ( x, y , z ) = p0 cos ⎜ ⎟ cos ⎜ ⎟ cos ⎜⎜ ⎟ , m, n = 0,1, 2,... ⎟ ⎝ Lz ⎠ ⎝ Lx ⎠ ⎝ Ly ⎠
[5. 47]
A few other comments about these results are useful. First, the physical relevance of the particular case where the three indices are zero must be discussed. In terms of displacements, the solution ω = k x = k y = k z = 0 would describe the free translations of an incompressible fluid. As the walls are assumed to be fixed, this kind of solutions is physically irrelevant. However, if the dual point of view is adopted, the particular case = m = n = 0 can be physically interpreted as a mode of uniform pressurization of the enclosure. Then, the plane waves studied in Chapter 4 are recovered as a special case corresponding to a single index differing from zero. They correspond to the longitudinal or axial modes. The tangential modes are such that a single index is zero and finally the oblique modes are such that the three indices differ from zero. According to the formula [5.46] it is found that the ratios between the relative amplitude of the Cartesian components of the modal vector ϕ , m, n , have specific values, which points out that the three directions are coupled together. On the other hand, it is also appropriate to pay attention to the modal density because the point is of crucial importance for computational and practical applications in the so-called “room acoustics” problems. A short look at
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367
formula [5.45] is sufficient to realize that the number of acoustic modes within a given frequency interval is much higher than in the case of two dimensional structures. This is a mere consequence of shifting from a 2D to a 3D continuous medium. To be more specific, let N (ω ) denote the number of modes whose circular frequencies are less than ω. Its value can be determined starting from the relation [5.45] written in the following equivalent form: ω2,m,n c 2f
=
k2,m,n
2⎞ ⎛ ⎛ ⎞2 ⎛ ⎞2 ⎛ n ⎞ ⎟ m ⎜ =π ⎜ ⎟ +⎜ ⎟ +⎜ ⎟ ⎜ Ly ⎟ ⎜ Lx Lz ⎟ ⎝⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎠ 2
[5.48]
Figure 5.4. Wave number lattice diagram
Relation [5.48] can be conveniently vizualised by means of the so-called wave number lattice diagram, see Figure 5.4. In this diagram, each individual mode is represented by a point defined by the corresponding wave number vector k,m ,n . N (ω ) is the total number of points of the lattice diagram that lie in or at the
boundaries of the first octant k x ≥ 0, k y ≥ 0, k z ≥ 0 , at a radial distance less than k = ω / c f of the origin of the axes. Detailed calculation was performed by [MAA
39] and can be found in particular in [PIE 91]. At high frequency, N (ω ) is approximated by the density of the oblique modes which outnumbers that of the axial and tangential modes by a large amount. The density of the oblique modes is given approximatively by the volume in an octant of the sphere with radius ω / c f divided by the “volume” V1 of an elementary cell (see right part of Figure 5.4). 4π ⎛ ω N (ω ) ⎜ 3 × 8 ⎜⎝ c f
3
⎞ ⎛ 1 ⎞ Lx Ly Lz ⎟⎟ × ⎜ ⎟ = 6π 2 ⎠ ⎝V1 ⎠
⎛ω ⎜⎜ ⎝ cf
⎞ ⎟⎟ ⎠
3
[5.49]
368
Fluid-structure interaction
The modal density, defined as the number of modes per unit frequency bandwidth, is thus approximated by: 2 dN 4π Lx Ly Lz f df c 3f
[5.50]
Therefore, in contrast with modal density of the 1D sound waves which is constant and proportional to the tube length, that of a rectangular box increases in proportion to the frequency squared and to the volume of the box. As the modal density increases, the modal synthesis approach to solve ‘room acoustics’ problems becomes less attractive and even rapidly unfeasible. This for the two following reasons. On one hand, the number of modes becomes impressively high as frequency and/or the room size is increased, requiring correlatively an overwhelmingly large finite element model for computation. On the other hand, the response properties, characterized typically by transfer functions, are shaped as a series of peaks of resonance so closely clustered to each other that their individual contribution to the response is completely blurred out in a continuum, as the result of the spectral broadening effect associated with damping. It is recalled that if the damping ratio ς n is small enough, the magnitude of an individual peak of resonant response is reduced by half when frequency is shifted from the resonant value f n by about ± f nς n , cf. [AXI 04] for instance. In qualitative agreement with the Schroeder criterium, assuming that a smoothed-out continuum spectral response is achieved if the average spacing of modal frequencies δ f mode becomes less than one third of f nς n , we arrive at the result that the acoustic response of the room loses completely its resonant behaviour at frequencies higher than the threshold value given by the condition: 1 fς δ f mode = δ f resonance n n 3 3
[5.51]
Substituting the reciprocal of [5.50] for δ f mode , the cut-off frequency between the modal and the continuum ranges of acoustic response is approximated by: 1/ 3
⎛ ⎞ 3 fc = c f ⎜ ⎜ 4π L L L ς ⎟⎟ x y z n ⎠ ⎝
[5.52]
A numerical application will help one grasp the practical implications of the theoretical results presented just above. For example, even in a fairly small living room Lx = Ly = 5m , Lz = 2.5m , adopting c f = 340 m/s , the density becomes higher than 20 modes per Hz at frequencies higher than 1 kHz (wavelength of 34 cm). The number of modes in the frequency range lower than 5 kHz is of the order of one million reaching about half a billion at 20 kHz. On the other hand,
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369
assuming a damping ratio of one percent, the resonant behaviour of the room disappears at frequencies higher than about 250 Hz. These introductive considerations clearly point out the severe limitations of the modal analysis methods to deal with room’s acoustic problems, as in most cases the frequency range below the Schroeder cut-off frequency is far too low to cover the most efficient part of the audiofrequency range. 5.1.2.2 Circular cylindrical enclosure
Figure 5.5. Circular cylindrical enclosure
The enclosure sketched in Figure 5.5 is defined as a cavity bounded by the walls of a cylinder of revolution closed at the top and bottom. Once more the boundary is supposed to be fixed. The modal problem is is written in cylindrical coordinates as: ∂ 2p 1∂ p 1 ∂ 2p ∂ 2p ⎛ ω + + + +⎜ ∂ r 2 r ∂ r r 2 ∂ θ 2 ∂ z 2 ⎜⎝ c f ∂p ∂p = ∂z 0 ∂z
H
∂p = ∂r
2
⎞ ⎟⎟ p = 0 ⎠
[5.53]
=0 R
Since the axial and radial boundary conditions are expressed as partial derivatives with respect to the corresponding coordinate only, we are encouraged to try once more the method of separation of the variables. Hence the pressure field is written as: p ( r,θ , z ) = A ( r ) B (θ ) C ( z )
The modal equations [5.53] are thus transformed into:
[5.54]
370
Fluid-structure interaction 2
1 ⎡ d 2 A 1 dA ⎤ 1 1 d 2 B 1 d 2C ⎛ ω ⎞ + + + +⎜ ⎟ =0 A ⎢⎣ dr 2 r dr ⎥⎦ r 2 B dθ 2 C dz 2 ⎜⎝ c f ⎟⎠ dC dC dA = = =0 dz 0 dz H dr R
[5.55]
The axial mode shapes are governed by: d 2C + k z2C = 0 dz 2
dC dC = dz 0 dz
;
=0
[5.56]
H
Whence the many infinite solutions: C ( z ) = cos
π z H
= 0,1, 2, …
[5.57]
Thus, as must be expected, the axial mode shapes of the cylindrical cavity identify with those of the plane waves in a tube of constant cross-section closed at both ends. Then, substituting the solution [5.57] into the first equation [5.55] multiplied by r 2 , one obtains: r 2 ⎡ d 2 A 1 dA ⎤ 1 d 2 B ⎛⎜ ⎛ ω + + + ⎜ A ⎢⎣ dr 2 r dr ⎥⎦ B dθ 2 ⎜ ⎜⎝ c f ⎝
2
⎞ ⎛ π ⎞2 ⎞ 2 ⎟⎟ − ⎜ ⎟ ⎟ r = 0 ⎠ ⎝ H ⎠ ⎟⎠
[5.58]
The circumferential, or azimuthal, mode shapes are governed by: 1 d 2B = constant ; B ( 0 ) = B ( 2π ) (2π periodic) B dθ 2
[5.59]
Whence the infinite many solutions: ⎧cos mθ Bm (θ ) = ⎨ ⎩ sin mθ
m = 0,1,2,...
[5.60]
This kind of solution has been already discussed in detail in [AXI 05] within the context of the natural modes of vibration of circular cylindrical shells. It is recalled that they arise as a mere consequence of the 2π periodicity of any field variable in cylindrical geometry. The circumferential index m = 0 characterises the breathing modes of the fluid contained within the cavity. Finally, substituting the solutions [5.60] into equation [5.58], the ordinary equation governing the radial shapes is obtained as: ⎡⎛ ω d 2 A 1 dA + + A ⎢⎜ 2 dr r dr ⎢⎜⎝ c f ⎣
2
⎞ ⎛ π ⎞2 m 2 ⎤ − − 2 ⎥=0 ⎟⎟ ⎜⎝ H ⎟⎠ r ⎥ ⎠ ⎦
;
dA dr
R
=0
[5.61]
3D Sound waves
371
With the aid of the variable transformation u = α r / R , where: ⎡⎛ ω α 2 = ⎢⎜ ⎢⎜⎝ c f ⎣
2 ⎞ ⎛ π ⎞2 ⎤ 2 ⎟⎟ − ⎜ ⎟ ⎥ R ⎠ ⎝ H ⎠ ⎥⎦
[5.62]
the differential equation [5.61] takes on the canonical form of the Bessel equation of integral order m: ⎡ m2 ⎤ d 2 A 1 dA + + A ⎢1 − 2 ⎥ = 0 2 du u du ⎣ u ⎦
[5.63]
Accordingly, the general solution of equation [5.61] is: ⎛ αr ⎞ ⎛ αr ⎞ Am ( r ) = am J m ⎜ ⎟ + bmYm ⎜ ⎟ ⎝ R ⎠ ⎝ R ⎠
[5.64]
where J m and Ym are the Bessel functions of order m, and of the first and the second kind respectively. However, it is clear that in the present problem bm necessarily vanishes since pressure must remain finite on the axis of the cylinder (see Appendix A4). On the other hand, the admissible values of the factor α are specified by the radial boundary condition: dAm ( r ) a α = m J m′ (α ) = 0 dr r = R R
[5.65]
Whence the radial mode shapes: ⎛α r ⎞ Am ,n ( r ) = J m ⎜ m ,n ⎟ ⎝ R ⎠
[5.66]
where α m ,n stands for the n-th root of the transcendental equation J m′ (u ) = 0 , see Table 5.1. Substitution of α m ,n into equation [5.62] yields the natural pulsations of the cylindical cavity by the condition: 2
2 ⎛ α m , n ⎞ ⎛ π ⎞ ⎛ ω , m , n ⎞ ⎜ R ⎟ + ⎜ H ⎟ = ⎜⎜ c ⎟⎟ ⎝ ⎠ ⎝ ⎠ ⎝ f ⎠
2
[5.67]
Gathering the partial results [5.57], [5.60], [5.66] and [5.67], the natural frequencies and pressure mode shapes of the cylindrical cavity are as follows:
372
Fluid-structure interaction
f ,m ,n =
cf 2π
2
⎛ π ⎞ ⎛ α m ,n ⎞ ⎜ ⎟ +⎜ ⎟ ⎝H⎠ ⎝ R ⎠
2
[5.68]
π z ⎧cos mθ ⎫ ⎛α r ⎞ ϕ ,m ,n ( r, θ , z ) = J m ⎜ m ,n ⎟ cos ⎨ ⎬ H ⎩ sin mθ ⎭ ⎝ R ⎠ ( p)
= 0,1,2,... is the axial rank of the mode, which specifies the number of pressure nodal planes perpendicular to the cylinder axis.
m = 0,1,2,... is the circumferential, or azimuthal rank of the mode, which specifies the number of pressure nodal diametral planes. n = 0,1,2,... is the radial rank of the mode, which specifies the number of pressure nodal cylinders, which of course are coaxial (axis Oz). The modes = m = 0 are the breathing modes which are also uniform in the axial direction. The modes n = 0 and m = 0 are the axial modes of the plane wave theory. Table 5.1. First roots of the radial characteristic equation J m′ (α m ,n ) = 0 n
m=0
1
0 3.8317 7.0156 10.1735 13.3237 16.4706
2 3 4 5 6
m=1 1.8412 5.3314 8.5363 11.7060 14.8636 18.0155
m=2 3.0542 6.0761 9.9695 13.1704 16.3475 19.5129
m= 3 4.2012 8.0152 11.3459 14.5859 17.7888 20.9724
In Table 5.1, the few first roots α m ,n of the characteristic radial equation are reported. The important point is that as the rank n of the roots is sufficiently high the characteristic equation tends asymptotically towards the trigonometric equation: ⎛ ( 2m − 1) ⎞ cos ⎜ α n ,m − π⎟=0 4 ⎝ ⎠
[5.69]
Equation [5.69] can be easily derived by using the formula [A4-16] in Appendix A4. As an interesting consequence, it is thus found that the modal density of the acoustical modes in a cylindrical enclosure is of the same order of magnitude as in a rectangular enclosure. Thus, the results [5.49] to [5.52] can be extended to the case of the present geometry simply by using the expression of the volume
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373
which is appropriate to the geometry of the box, or cavity. To proceed in the qualitative description of the acoustical modes in cylindrical geometry, we adopt the following numerical values H = 10 m, R = 5 m and we assume that the enclosure is filled with air at STP conditions, ρ f 1.2 kgm -3 , c f 344 ms-1 . The natural frequency of the first longitudinal mode is f1,0,1 = c f /2H = 17.2 Hz and the lowest frequency of the non-plane modes is f1,1,1 39.87 Hz . Such a numerical result indicates that in a cylindrical enclosure of rather low aspect ratio η = H / R = 2 , the spectral range of validity of the plane wave model is very narrow. It increases proportionally to η as indicated by the frequency formula [5.68] which implies: f ,m ,n
2
cf
f ⎛α H ⎞ ⎛α H ⎞ = 1 + ⎜ m , n ⎟ ⇒ ,m , n = 1 + ⎜ m , n ⎟ f ,0,1 2H ⎝ π R ⎠ ⎝ π R ⎠
2
[5.70]
In the present example, if the radius is changed into R = 5 cm , f1,0,1 remains the same while the lowest frequency of the non-plane mode becomes f1,1,1 = 2016 Hz . The spectral range of validity of the plane wave model will be discussed from the viewpoint of travelling guided waves in subsection 5.2. Coming back to the case of the first cylinder of low aspect ratio η = 2 , Figures 5.6 and 5.7 show a few pressure mode shapes. In the right hand-side of these figures, the radial profile of the modal pressure is plotted versus the reduced distance r/R from the cylinder axis. In the left hand-side, the corresponding cross-sectional pressure mode shapes are vizualized. Figure 5.6 refer to the breathing modes and Figure 5.7 to the m ≠ 0 modes. Orthogonality of the mode shapes of distinct circumferential or longitudinal indices is obvious, as it reduces to that of the ordinary circular sine and cosine functions. However, in the case of distinct radial indices m orthogonality is less obvious and worthy of a comment. Actually, it results from a property of the Bessel functions known as the Lommel integrals, which are presented here without mathematical proof:
∫
R
∫
R
0
0
2
⎫ ⎛ R 2 ⎪⎧ n2 ⎞ ⎛ ⎛ αr ⎞⎞ 2 2⎪ ⎜ J n ⎜ R ⎟ ⎟ rdr = 2 ⎨( J n′ (α ) ) + ⎜ 1 − α 2 ⎟ ( J n (α ) ) ⎬ ⎝ ⎝ ⎠⎠ ⎝ ⎠ ⎩⎪ ⎭⎪
J n (α1 )J n (α 2 )rdr =
R2 {α 2 J n (α1 ) J n′ (α 2 ) − α1 J n (α 2 ) J n′ (α1 )} α − α 22 2 1
[5.71]
[5.72]
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Figure 5.6. Pressure mode shapes of a few breathing modes: plots of the radial profile and cross-sectional visualization
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375
Figure 5.7. Pressure mode shapes of a few m ≠ 0 modes, plots of the radial profile (meridian plane θ = 0°) and cross-sectional visualization
In the present context, the Lommel integrals are particularized to the case α1 = α m ,n1 , α 2 = α m ,n2 where m is fixed and the radial indices n1 and n2 are distinct. Since the cylindrical wall of the enclosure is assumed to be fixed, the relation J m′ (α1 ) = J m′ (α 2 ) = 0 holds. Substitution into the integral [5.72] gives immediatly the expected result of orthogonality between two distinct radial mode shapes, in full agreement with the general result [5.23]. The same would be true for any conservative boundary condition. In particular, the case of a pressure node at the cylindrical wall implies the relation J m (α1 ) = J m (α 2 ) = 0 , where α1 = β m ,n1 ,
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α 2 = β m ,n 2 and β m ,n designates the n-th root of the transcendental equation J m ( u ) = 0 . Substitution into the integral [5.72] gives once more the expected result.
The orthogonality of the Bessel functions allows one to expand any function f(x) defined on the interval [0,1] as a series of Bessel functions: ∞
f ( x) = ∑ a j J m ( βm, j x)
[5.73]
j =1
Here again β m , j is the j-th root of J m ( x ) = 0 . The coefficients of the series are: 1
aj =
2 ∫ f ( x ) J m ( β m , j x ) xdx 0
( J ′ (β )) m
[5.74]
2
m, j
This kind of series if of course similar to the Fourier series and are sometimes termed Bessel-Fourier series. EXAMPLE.– Expansion of the rectangular pulse f ( x ) = 1 ; x ∈ [ 0,1] as a Bessel series.
af
To expand the function f x = 1 in Bessel series on the interval 0,1 , one is naturally led to select the Bessel function of zero order J 0 since all the other Bessel functions of the first kind vanish at x = 0, while all the Bessel functions of the second kind tend to infinity. Thus the series is written as: ∞
f ( x ) = ∑ a j J 0 ( β 0, j x )
[5.75]
j =1
With the aid of formula [A4.8] in Appendix A4, it follows that: x
⌠ ⎮ ⎮ ⌡0
zJ 0 ( z ) dz = xJ 1 ( x ) = − xJ 0′ ( x )
[5.76]
Whence: 1
aj =
2 ∫ J 0 ( β 0, j x ) xdx 0
( J ′(β )) 0
2
0, j
1
=
2 ∫ xλ0, j J 0 ( β 0, j x )d ( β 0, j x ) 0
2 0, j
λ
( J (β )) 1
0, j
2
=
2 β 0, j J 1 ( β 0, j )
[5.77]
Substitution in relation [5.75] yields the final form of the series: ∞
1=2∑ j=1
J 0 ( β 0, j x )
β 0, j J 1 ( β 0, j )
[5.78]
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377
The series truncated up to the order 20 is shown in Figure 5.8. Its shape is distinct from that of a Fourier series expansion of the same function, accuracy of the latter being better according to the least square criterion. However, the area under the curve can be reasonably well reproduced even if the truncation order is fairly low.
Figure 5.8. Rectangular pulse expanded in series of Bessel function of the first kind and of order 0
Figure 5.9. Pie shaped cylindrical sector
To conclude on the analytical determination of acoustical modes in cylindrical geometry, it is of interest to notice that the solution established for a cylindrical enclosure of revolution can be easily transposed to the case of a circular cylindrical sector, that is a cylinder with pie-shaped cross-sections, as shown in Figure 5.9. Actually, the variable separation method still applies, leading to the same general
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solutions for the functions A ( r ) , B (θ ) and C ( z ) . However, major differences occur as far as the mode shapes are concerned because of the boundary conditions at the planes bounding the cylindrical sector. The circumferential functions B (θ ) are still found to be a linear form of cos mθ and sin mθ ; however, they are no longer constrained to be 2π periodic and m is generally not an integral number. Let us assume for instance the fixed conditions: ∂p ∂p = =0 ∂θ θ = 0 ∂θ θ =θ
[5.79]
0
It follows that the circumferential mode shapes are necessarily: Bm (θ ) = cos mθ , where m =
kπ θ0
k = 0,1, 2..
[5.80]
The pressure mode shapes follow as: π z ⎛α r ⎞ p cos mθ ϕ (,m),n ( r, θ , z ) = J m ⎜ m ,n ⎟ cos H ⎝ R ⎠
[5.81]
where again α m ,n is the n-th root of the Bessel function of the first kind of real index m. For the free conditions, p θ = 0 = p θ =θ = 0 0
[5.82]
the pressure mode shapes are: π z ⎛α r ⎞ p ϕ (,m),n ( r, θ , z ) = J m ⎜ m ,n ⎟ cos sin mθ R H ⎝ ⎠ ( 2k +1) π k = 0,1, 2,... where m = 2θ0
[5.83]
In both cases the natural frequencies have the same formal expression but the numerical values differ since the circumferential index is changed and so also the roots α m ,n . 5.1.2.3 Spherical enclosure Definition of the spherical coordinates and notations used in this book are recalled in Figure 5.10. The modal problem is expressed in spherical coordinates as (cf. equation [2.238]) :
3D Sound waves
∂ 2 p 2 ∂p ∂p 1 ∂ 2 p ∂2 p ⎛ ω 1 1 + + 2 + 2 + +⎜ 2 2 2 2 ∂r r ∂r r tan ϕ ∂ϕ r ∂ϕ ( r sin ϕ ) ∂θ ⎜⎝ c f
+ ( C.B.C. )
379
2
⎞ ⎟⎟ p = 0 ⎠
[5.84]
As in the preceding problems, presentation is restricted here to the standard boundary conditions of either a fixed or a free boundary. Once more, separation of variables leads to an analytical solution in terms of special functions, namely the spherical Bessel functions and the Legendre polynomials. The subject is treated in detail in many textbooks devoted to applied mathematics and theoretical acoustics, see for instance [ANG 61], [STA 70], [BLA 00] and a brief description of these functions will be given here.
Figure 5.10. Spherical geometry
The spherical coordinates are related to the Cartesian coordinates as shown in Figure 5.10, where θ denotes the longitude and ϕ the colatitude. The pressure field is written as: p ( r,θ , z ) = A ( r ) B (θ ) C (ϕ )
[5.85]
differential equation [5.84] is thus transformed into the form: 2
1 ⎛ 2 d2A dA ⎞ 1 1 d 2 B 1 ⎛ d 2C 1 dC ⎞ ⎛ ω r ⎞ r 2 r + + + ⎜ 2+ ⎜ ⎟ ⎟+⎜ ⎟ = 0 2 2 2 A ⎝ dr dr ⎠ ( sin ϕ ) B dθ C ⎝ dϕ tan ϕ dϕ ⎠ ⎜⎝ c f ⎟⎠
[5.86]
Depending whether the surface of the sphere is fixed or free, the boundary condition dA is = 0 , or A ( R ) = 0 respectively. The radial part of [5.86] can be separated dr r = R to produce the ordinary differential equation:
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1 ⎛ 2 d2A dA ⎞ ⎛ ω r ⎞ + 2r ⎟ + ⎜ ⎟ = α 2 ⎜r 2 A ⎝ dr dr ⎠ ⎜⎝ c f ⎟⎠
[5.87]
dA = 0, or A ( R ) = 0 dr r = R
where the separation constant is denoted α 2 . On the other hand, noticing that again B (θ ) is necessarily 2π periodic, the azimuthal solutions are found to be: ⎧cos mθ Bm (θ ) = ⎨ where m = 0,1, 2,.. ⎩ sin mθ
[5.88]
Substituting the radial equation [5.87] and the solutions [5.88] into [5.86] the following ordinary differential equation is obtained: ⎛ d 2C 1 dC ⎞ ⎛ 2 m2 ⎞ ⎜ 2+ ⎟ + ⎜α − 2 ⎟ C = 0 tan ϕ dϕ ⎠ ⎝ sin ϕ ⎠ ⎝ dϕ
[5.89]
With the aid of the transformation z = cos ϕ , equation [5.89] takes on the canonical form of the so called associated Legendre equation: 2
(1 − z ) ddz f 2
2
− 2z
df ⎛ μ ⎞ + ⎜ν (ν + 1) − ⎟ f =0 dz ⎝ 1 − z2 ⎠
[5.90]
where ν and μ are two real constants. In the particular case μ = 0, the solutions of [5.90] are in terms of the so called Legendre or spherical functions. If μ differs from zero, the solutions are in terms of the associated Legendre functions. For mathematical convenience, the case μ = 0 is described first. The solution of the Legendre equation is outlined in Appendix A5. The important point to retain here is that it tends to infinity on the polar axis z = ±1 , unless the separation constant can be written as: α 2 = n ( n + 1)
[5.91]
where n is an integral number. Clearly, in the present problem, to be physically admissible the solution of [5.90] must comply with the condition [5.91]. Incidentally, the way discrete values of the separation constant enter into the present problem differs from the way discrete values of frequency, or wavenumber, enter into a modal problem. Here, the sequence n = 1,2,… is prescribed to avoid unphysical mathematical solutions, whereas in a modal probem it would be introduced through reflective boundary conditions to obtain nontrivial mathematical solutions. The admissible solutions are
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381
the Legendre polynomials, denoted Pn ( z ) = Pn ( cos ϕ ) which may be defined for instance by Rodrigues’s formula: Pn ( z ) =
( −1)
n
2n n !
d n ( z 2 − 1)
n
d zn
[5.92]
The Legendre polynomials are orthogonal over the interval [-1,1]: 1
⌠ ⎮ P n1 ⎮ ⌡−1
⎧ 2 n =n =n ( z ) Pn2 ( z ) dz = ⎪⎨ 2n + 1 1 2 ⎪⎩ 0 otherwise
[5.93]
Coming back to the radial equation [5.87], using the admissibility condition [5.91], it can be transformed into: d 2 A 2 dA ⎛ 2 n ( n + 1) ⎞ + +⎜k − ⎟A=0 dr 2 r dr ⎝ r2 ⎠
[5.94]
dA = 0, or A ( R ) = 0 dr r = R
It turns out that the differential equation [5.94] can be transformed into a Bessel equation, by changing A ( r ) into: U (r) =
A(r)
[5.95]
r
Actually, substitution of [5.95] into [5.94] yields the boundary value problem expressed in terms of the Bessel equation: 2 d 2U 1 dU ⎛ 2 ( n + 1/ 2 ) ⎞ k + + − ⎜ ⎟U = 0 ⎟ dr 2 r dr ⎜⎝ r2 ⎠ dU U r + = 0, or U ( R ) = 0 dr 2 r = R
[5.96]
The solutions of the radial equation [5.94] are thus of the physically admissible form: An ( r ) =
J n +1/ 2 ( kr ) kr
= jn ( kr )
[5.97]
382
where
Fluid-structure interaction
jn ( x ) is the spherical Bessel function of integral index n already
encountered in Chapter 4, subsection 4.2.1.8 in the context of Bessel horns. Table 5.2 gives the first zeros of jn ( x ) and Table 5.3 those of their first derivative. Table 5.2. First roots xn of jn ( x ) = 0
n=0
1
π 2π 3π 4π 5π
2 3 4 5
n=1 4.493 7.725 10.90 14.07 17.22
n=2 5.763 9.095 12.32 15.52 18.69
n= 3 6.988 10.41 13.70 16.92 20.12
Table 5.3. First roots xn′ of jn′ ( x ) = 0
n=0
1
0 4.493 7.725 10.90 14.07
2 3 4 5
n=1 2.082 5.940 9.206 12.41 15.58
n=2 3.342 7.290 10.614 13.85 17.04
n= 3 4.514 8.578 11.97 15.25 18.47
The acoustic modes of the spherical enclosure which are symmetric about the polar axis (independent of θ ) are characterized by the natural frequencies and pressure mode shapes: f n , =
ϖ n , c f 2π R
⎛ϖ r ⎞ p ; ϕ n( ,) ( r, z ) = Pn ( z ) jn ⎜ n , ⎟ ⎝ R ⎠
[5.98]
ϖ n is either equal to xn′ or xn , depending on whether the boundary is fixed or free. Figure 5.11 shows the first few axisymmetrical pressure mode shapes for a rigid sphere R = 1 m, filled with air at STP.
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Figure 5.11. Pressure mode shapes of spherical modes: axisymmetric family m = 0
383
384
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Figure 5.12. Meridian polar plots of the pressure mode shapes
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385
The left-hand side plots refer to the radial mode shape (spherical Bessel functions complying with the fixed boundary condition at r = R) and the polar plots in the right-hand side show the pressure field (dashed line) along a meridian of the sphere (full line). The mode shapes corresponding to even n values are symmetrical about the equatorial plane, whereas those corresponding to odd n values are antisymmetrical about the equatorial plane. At the sphere origin, the pressure of the breathing modes n = 0 has an extremum, hence particle velocity is zero, as should be expected owing to the central symmetry of the system. The first breathing mode at zero frequency is not represented in Figure 5.11. It corresponds to the constant pressure mode already encountered in Cartesian and cylindrical geometries. A few three dimensional visualizations are also displayed in colour plates 5 and 6. The most general type of spherical modes correspond to m, hence μ, distinct from zero. The solution of the associated Legendre equation is outlined in Appendix A5. Again, it tends to infinity on the polar axis z = ±1 , unless the separation constant can be written as α 2 = n ( n + 1)
[5.99]
Hence, the radial equation is the same as in the case of the axisymmetric modes and the solutions are once more the spherical Bessel functions of the first kind, which are constrained to comply with the same radial boundary condition as in the symmetrical case. Furthermore the C (ϕ ) equation is of the type [5.90] with ν = n and μ = m 2 . The solutions are in terms of the associated Legendre functions: Cn ,m ( z = cos ϕ ) = Pnm ( z ) = (1 − z 2 )
m/2
d m Pn ( z ) dz m
[5.100]
Substituting [5.92] into [5.100] yields: m n
P
(z) =
( −1)
m
(1 − z )
2 m/2
2n n !
d m + n ( z 2 − 1)
n
[5.101]
dz m + n
Accordingly, if m is greater than n, the derivative in [5.101] is identically zero. On the other hand, in the above formula, the index m+n cannot be negative as it specifies the number of times a quantity is differentiated. Therefore, the integral index m must be selected within the range: −n ≤ m ≤ n
[5.102]
Furthermore, it turns out that by changing the sign of m, Pn( ) is changed by a multiplicative constant only, so it is sufficient to consider the non negative values of m. On the other hand, the natural frequency is found to be independent of m, since it is specified by using the radial boundary condition, which is the same as in the m
386
Fluid-structure interaction
symmetrical case (m = 0). Therefore, the natural frequencies and pressure mode shapes of the spherical standing waves follow as: ϖ n , c f
⎛ ϖ r ⎞ ⎧cos mθ ⎫ p ; ϕ n( ,m) , ( r , z ) = Pnm ( z ) jn ⎜ n , ⎟ ⎨ ⎬ 2π R ⎝ R ⎠ ⎩ sin mθ ⎭ n = 0,1, 2,... ; 0 ≤ m ≤ n ; = 1, 2,... f n ,m , =
[5.103]
Again ϖ n , is either equal to xn′ or xn depending on whether the boundary is fixed or free. As expected, the associated Legendre polynomials are orthogonal over the interval [-1,1], complying with the relations: 1
⌠ ⎮ Pm n1 ⎮ ⌡−1
⎧ 2 ( n + m )! n =n =n ( z ) P ( z ) dz = ⎪⎨ ( 2n + 1)( n − m )! 1 2 ⎪ 0 otherwise ⎩ m n2
[5.104]
Figure 5.12 shows a few pressure mode shapes represented as polar plots for a rigid sphere R = 1 m, filled with air at STP, see also colour plate 7. NOTE – Sphere bounded by a pressure nodal surface p ( r = R,θ , ϕ ) = 0 If pressure vanishes at the surface of the sphere, the breathing modes have the following natural frequencies and pressure mode shapes: f 0,0, =
c f 2R
⎛ π r ⎞ sin ( π r / R ) ( p) ; ϕ 0,0, ; = 1, 2,... ( r , z ) = j0 ⎜ ⎟= π r / R ⎝ R ⎠
[5.105]
At the centre of the sphere, the pressure is maximum and the radial particle velocity vanishes. However, as in the case of the cylindrical geometry, the converse is not true, that is the pressure nodes do not necessarily coincide with the particle velocity extrema. This is again merely a consequence of the mass and momentum equations written in spherical coordinates as: ⎛ u du ⎞ iω p + ρ f c 2f ⎜ + ⎟ = 0 ⎝ r dr ⎠ dp =0 iωρ f u + dr
[5.106]
For the modes n > 0, the pressure vanishes at the center of the sphere. Before leaving the subject, it is also of interest to mention that spherical coordinates are also well suited to deal with conical and wedge geometries. If the surface is conical, it is appropriate to orient the cone so that its axis coincides with the polar axis Oz of the spherical coordinate system. If the conical surface is fixed, the boundary condition is simply ∂p / ∂ϕ = 0 . In the case of a wedge, the polar axis Oz is chosen along the intersection of the planes limiting the wedge. Denoting their
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387
polar angles θ1 and θ 2 , the boundary condition for a fixed wedge is simply ∂p / ∂θ = 0 at θ1 and θ 2 . 5.2. Guided wave modes and plane wave approximation 5.2.1
Introduction
A wave can be guided along certain directions by restricting the space of propagation by reflecting boundaries in the other directions. As a consequence, the nature of a guided wave is modified in comparison with the unbounded case. As desired, it is progressive in the direction of transmission and stationary in the other directions. Sound waves propagating in a tube are a typical example which was already described in Chapter 4 based on the plane wave model. Here, the purpose is to extend the analysis beyond the domain of validity of the plane wave model, that is in the range of wavelengths less than the transverse dimensions of the tube. Actually, the concept of guided waves has been already introduced in [AXI 05] Chapter 1 in the context of the elastodynamics in solids. The purpose there was to point out the usefulness of modelling solid bodies as structural elements endowed with geometrical particularities which allows one to reduce the dimension of the problem by using a 2D, or even a 1D equivalent continuous medium instead of the actual 3D medium. It is worthwhile recalling here that the analysis of the guided propagation of elastic waves in a solid is made particularily arduous due to the concomitant presence of dilatational and shear waves which interact with each other through complicated laws of wave reflexion at the surface of the solid. As a consequence, the major features of the guided wave propagation in an elastic solid were discussed, based on a particular example of academic interest, especially selected for its physical and mathematical simplicity. In the present case, the problem is greatly simplified because in an inviscid fluid no shear waves exist and the dilatational waves are found to be reflected by a fixed wall according to the ordinary specular law. Indeed, by virtue of the energy and momentum equation, the normal component of the particle velocity must change sign due to the reflection, whereas the tangential component is left unchanged. Here, the object is not to repeat the presentation made in [AXI 05] but to take advantage of the simplifications which hold in an inviscid fluid to treat the problem in a more systematical way than in the case of solids. The general principle of the method consists of introducing the concept of guided wave modes. A guided wave mode can be formally defined as the nontrivial harmonic solutions of the homogeneous wave guide problem. As could be expected, they are indefinitely many such solutions which have the features of standing waves in the stationary directions, thereafter referred to as transverse, and of progressive backward and forward waves along the direction of transmission, thereafter referred to as longitudinal, or axial. Due to the standing nature of the guided wave in the transverse directions, they can be conveniently individualized by two integral
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Fluid-structure interaction
indices, which serve to specify the rank of the guided wave mode. They serve also to individualize the branch of the associated dispersion equation which relates the phase velocity of the progressive part of the mode to the frequency, or equivalently the frequency and the wave number. 5.2.2
Rectangular waveguides
5.2.2.1 Guided mode waves
Figure 5.13. Rectangular waveguide
The waveguide is shown in Figure 5.13. Assuming it extends infinitely far in the Ox direction and the lateral walls are rigid and fixed, the harmonic waves are governed by the following boundary value problem: ∂2p ∂2p ∂2p + + + k2 p = 0 ∂ x2 ∂ y2 ∂ z2 ∂p ∂y
y =− b
∂p = ∂y
y =+ b
∂p = ∂z
z =− a
∂p = ∂z
[5.107] =0 z =+ a
In line with the procedure followed in a modal problem, we search for nontrivial solutions of the general type: ϕ m( ,n ) ( x, y , z ) eiω t pg
[5.108] ( pg )
indicates that we have to deal with the (normalized) where the superscript pressure field of a guided wave mode. On the other hand, two lateral indices denoted m and n are used instead of three since no wave reflection occurs along the longitudinal direction. The problem [5.107] is identical to the modal system [5.39] and [5.40], except for the longitudinal boundary conditions and the coordinate system which are not the same. It follows that the guided waves are necessarily of the type:
3D Sound waves
mπ ( b + y ) nπ ( a + z ) cos pm ,n ( x ) eiωt 2b 2a ; n = 0,1, 2,...
ϕ m( ,n ) ( x, y , z ) eiωt = cos pg
m = 0,1, 2,...
389
[5.109]
Substituting the solution [5.109] into equation [5.107] and performing the projection onto the transverse mode shapes yields the following ordinary differential equation which governs the guided wave of rank m,n: d 2 pm ,n dx 2
+ km2 ,n pm ,n = 0 ; m = 0,1, 2,...
; n = 0,1, 2,...
[5.110]
The associated wave numbers km ,n are found to comply with the following equation of dispersion: ⎛ω km2 ,n = ⎜ ⎜ cf ⎝
2
⎞ ⎧⎪⎛ mπ ⎞ 2 ⎛ nπ ⎞2 ⎫⎪ ⎟⎟ − ⎨⎜ ⎟ +⎜ ⎟ ⎬ ⎠ ⎩⎪⎝ 2b ⎠ ⎝ 2a ⎠ ⎭⎪
[5.111]
As should be expected, the general solution of the homogeneous equation [5.99] is a linear superposition of two harmonic waves travelling forward and backward respectively: pm,n ( x; ω ) eiω t = A+ e
i (ω t − km ,n x )
+ A− e
i (ω t + km , n x )
[5.112]
Hence, the pair of forward and backward guided waves of rank m,n are defined as: ⎛ mπ ( y + b ) ⎞ ⎛ nπ ( z + a ) ⎞ i(ω t − km ,n x ) pg + ϕ m( ,n ) ( x, y , z; ω ) eiω t = A+ cos ⎜ ⎟ cos ⎜ ⎟e 2b 2a ⎝ ⎠ ⎝ ⎠ ⎛ mπ ( y + b ) ⎞ ⎛ nπ ( z + a ) ⎞ i (ω t + km,n x ) pg − ϕ m( ,n ) ( x, y , z; ω ) eiω t = A− cos ⎜ ⎟ cos ⎜ ⎟e 2b 2a ⎝ ⎠ ⎝ ⎠
[5.113]
m = 0,1, 2,... ; n = 0,1, 2,...
Once more, the constants A+ and A− can be selected according to the norm condition desired. Since the transverse shapes of the guided wave modes are the same as those of the natural acoustic modes, they can be used as a vector basis to expand in series the solution of any waveguide forced problem, as illustrated later in subsection 5.3.3. The important point to be discussed now concerns the propagation properties of the guided wave modes in relation to their rank. Considering first the forward and backward modes of ranks m = n = 0, it is an easy task to check that they correspond precisely to the travelling plane waves already identified in Chapter 4. Such waves propagate at the constant phase velocity: ( ) = c0,0 ψ
ω = cf k00
[5.114]
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Fluid-structure interaction
As the guided wave modes of higher rank are concerned, propagation is more complicated. Equation [5.111] implies that the wave numbers are either real, or purely imaginary, depending whether the frequency is higher or lower than the cutc off value ω m( ,)n :
ωm( ,)n = c f c
⎧ ⎪k ( ± ) = ± ω ⎪ m ,n 2 2 cf ⎪ ⎛ mπ ⎞ ⎛ nπ ⎞ + ⇒ ⎨ ⎜ ⎟ ⎜ ⎟ ⎝ 2b ⎠ ⎝ 2a ⎠ ⎪ iω ⎪km( ±,n) = ± cf ⎪⎩
⎛ ω (c) 1 − ⎜ m ,n ⎜ ω ⎝ ⎛ ωm( c,)n ⎜ ⎜ ω ⎝
⎞ ⎟ ⎟ ⎠
2
if ω ≥ ωm( ,)n c
[5.115]
2
⎞ (c) ⎟ − 1 if ω ≤ ωm ,n ⎟ ⎠
Consequently, progressive waves are obtained in the high frequency range ω > ω m( ,)n c
solely. Above the cut-off value, lossless transmission occurs at the phase speed: cm( ,n) = ψ
ω = km ,n
±c f ⎛ ω (c) ⎞ 1 − ⎜ m ,n ⎟ ⎜ ω ⎟ ⎝ ⎠
[5.116]
2
Hence, the non-plane guided waves are dispersive, especially near the cut-off c frequency. In the low frequency range ω < ω m( ,)n , the solution −i km ,n is acceptable from the physical standpoint since it leads to an evanescent wave, which means that the wave amplitude decreases exponentially with the distance travelled: ⎛ mπ ( y + b ) ⎞ ⎛ nπ ( z + a ) ⎞ iωt − λm ,n (ω ) ( x, y, z; ω ) eiωt = A+ cos ⎜ ⎟ cos ⎜ ⎟e e 2b 2a ⎝ ⎠ ⎝ ⎠ x
ϕ m( ,n
pg + )
[5.117]
The characteristic length of exponential decrease is the real wavelength of the guided mode: λm ,n (ω ) =
cf
(ω ) (c)
m ,n
2
[5.118] −ω
2
Hence, the higher the frequency ratio ωm( ,)n / ω , the closer from the source the c
evanescent wave fades out. Of course, the other mathematical solution +i km ,n is to be rejected as it would imply a spatially amplified wave without any external supply of energy.
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391
Figure 5.14. Frequency spectrum of sound waves in waveguide of square cross-section
All the results presented in this subsection are in full agreement with those presented in [AXI 05] based on an academic example involving a single index. In the same manner, the dispersion equation [5.111] can be represented graphically as shown in Figure 5.14, which refers to a waveguide in air at STP and 2a = 2b = 1 m. In this kind of diagram, commonly called frequency spectrum, the wave frequency is plotted versus the real part and the imaginary part of the complex wave number km ,n . To help visualization, Real( km ,n ) is plotted as a positive abscissa whereas Im( km ,n ) is plotted as a negative abscissa. Furthermore, the travelling part of the branches which belong to Real( km ,n ) are plotted in full lines while the vanishing parts belonging to Im( km ,n ) are plotted in dashed lines. In such a diagram, to check whether a mode (m,n) can propagate or not at a given frequency, it suffices to draw the corresponding horizontal line and check whether the intersection occurs with the full part, or the dashed part of the branch line. Another way to present the information graphically is to plot the real part of the phase velocity versus frequency, as depicted in Figure 5.15. At the cut-off value, ψ cm( ,n) tends to infinity, except for the plane wave mode of course. As the frequency
392
Fluid-structure interaction
Figure 5.15. Phase velocity of the guided waves (m,n) as a function of frequency ( ) increases, cm( ,n) tends to the constant value c0,0 = c f and dispersion becomes ψ
ψ
negligible. The group velocity cm( ,n) is easily derived as: g
(g)
cm ,n
⎛ ω (c) ⎞ dω = = c f 1 − ⎜ m ,n ⎟ ⎜ ω ⎟ dkm ,n ⎝ ⎠
2
[5.119]
Then, as it should be expected, no wave energy is propagated at and below the cutoff frequency and group velocity tends to c f as frequency increases, see Figure 5.16. The major point of such dispersive properties is that, whatever the transverse shape of the waves emitted by the source may be, the waveguide filters out all the components which are emitted at frequencies below the cut-off values [5.115]. In particular, validity of the plane wave model presented in Chapter 4 is thus asserted in the low frequency range below the cut-off value of the first non-plane wave mode. f 0,1( ) or f1,0( ) = c
c
cf 2 LT
where LT denotes the largest transverse dimension of the waveguide.
[5.120]
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393
Figure 5.16. Phase velocity (full lines) and group velocities (dashed lines) of the guided waves (m,n) as a function of frequency
However, if the waveguide presents geometrical accidents, sudden changes in the cross-section, or in the direction of transmission for instance, near such places non-plane wave components exist, which extend over distances scaled by the wavelengths in agreement with formula [5.118]. The phenomenon is similar to the tunnelling effect mentioned in Chapter 4, in the context of sound propagation in horns and the Schrödinger equation, since in both cases we are in the presence of waves which decay progressively and not abruptly with distance. 5.2.2.2 Physical interpretation Finally, it may be worth adapting the physical interpretation of the dispersion equation already outlined in [AXI 05] Chapter 1 to the present case. The presentation follows that given in [BLA 00]. For the sake of simplicity, formula [5.113] is particularized to a guided mode of rank m,0 travelling along the longitudinal direction Ox (unit vector i ). Such a wave is then expressed as the combination of a pair of waves travelling also in the transverse direction Oy (unit vector j ): mπ ( y + b ) mπ ( y + b ) ⎛ ⎞ ⎛ ⎞ ⎛ mπ ( y + b ) ⎞ i (ω t − km ,0 x ) A+ ⎪⎧ i ⎝⎜⎜ ω t + 2 b − km ,0 x ⎠⎟⎟ i ⎝⎜⎜ ω t − 2 b − km ,0 x ⎠⎟⎟ ⎪⎫ = + A+ cos ⎜ e e e ⎨ ⎬ ⎟ 2b 2 ⎪ ⎝ ⎠ ⎪⎭ ⎩
[5.121]
394
Fluid-structure interaction
A few elementary manipulations suffice to rewrite them as a pair of plane longitudinal waves travelling at oblique incidence along the directions: ± = sin θ m i ± cos θ m j [5.122] The angle θ m between the directions j and ± is determined by transforming the argument of the exponentials as follows. Using the dispersion equation [5.111] we obtain: ⎛ ⎛ mπ ( y + b ) x = ω ⎜t − ⎜ ω t − km ,0 x ± ⎜ ⎜ cf 2b ⎝ ⎝
2
⎛ mπ c f ⎞ ( y + b ) ⎛ mπ c f 1− ⎜ ⎟ ± ⎜ c f ⎝ 2bω ⎝ 2bω ⎠
⎞ ⎞ ⎞⎟ ⎟ ⎟⎟⎟ ⎠⎠ ⎠
[5.123]
The angle θ m is defined as: ω ⎛ mπ ⎞ θ m = ± cos−1 ⎜ ⎟ ; k= 2 cf ⎝ bk ⎠
[5.124]
which allows one to write [5.123] as the argument of a plane wave of wavenumber k, travelling at the angle θ m with respect to the transverse Oy direction. The final form of [5.121] is expressed as: ⎛ mπ ( y + b ) ⎞ i (ω t − km ,0 x ) A+ ⎧ i (ω t − r .km( +,0) ) i (ω t − r .km( −,0) ) ⎫ A+ cos ⎜ = +e ⎨e ⎬ ⎟e 2b 2 ⎩ ⎭ ⎝ ⎠
[5.125]
The position vector and wave number vectors are: r = xi + ( y + b ) j ω km( +,0) = k + = ( sin θ m i + cos θ m j ) cf − ω km( ,0) = k − = ( sin θ m i − cos θ m j ) cf
[5.126]
Figure 5.17 shows a schematic view of the wave system taken at a given time t. It is pictured as the traces of a sequence of planes of constant phase. The direction of propagation ± is indicated by a white line which makes an angle θ m with the Oy + − direction. The km( ,0) waves are moving upward (full black lines) and km( ,0) waves are moving downward (dashed white lines).
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395
Figure 5.17. Guided waves as a system of two plane waves travelling at oblique incidence
The former are incident on the upper boundary (z = b) and are at the same time the waves reflected by the lower boundary (z = -b). The reverse holds concerning the downward waves. Where a wave plane intersects a boundary, a reflected wave is initiated in such a way that the boundary conditions are satisfied. To understand the physical meaning of the apparent phase speed we consider two planes of constant phase belonging to the same kind of waves and distant of one wavelength λ . The distance along the 0x direction is λx = λ / cos θ m . The time taken by a wave of a given frequency f to sweep this distance is the same whatever the corresponding value of θ m may be. Thus, the phase velocity of the guided wave mode corresponds to the apparent speed with which the wavefronts (crests or troughs) sweep along the Ox direction:
396
Fluid-structure interaction
c( ) =
cf ωλx ωλ = = = 2π 2π sin θ m sin θ m
a
cf ⎛ mπ c f ⎞ 1− ⎜ ⎟ ⎝ 2bω ⎠
2
=
ω ψ = cm( ,0) km ,0
[5.127]
The superposition of the two travelling waves creates a stationary wave in the Oy direction which must comply with the boundary conditions at y = ± b . Figure 5.17 corresponds to m = 3, that is three half transverse wavelengths. As indicated by the fourth expression of the phase velocity in [5.127], m being fixed, the phase velocity increases as frequency is lowered, simply because if ω decreases, λ = c f / ω increases, and so does θ m to keep the transverse trace wave length unchanged. Such a behaviour is clearly depicted in the colour plates 8 and 9. At normal incidence θ m = π / 2 , the phase velocity is infinite, which is a natural result since all the points of a same wavefront reach simultaneously any line parallel to the Ox direction. Furthermore, normal incidence corresponds to the cut-off frequency of the guide mode of rank (m,0). It can be verified that if the frequency is mπ c f c , there is no more possibility to accomodate the lowered futher than ωm( ,0) = 2b required number of half wavelengths in the 0y direction of the guide. As the apparent wavelength in the Oy direction is λ y = λ sin θ m , at normal incidence λ y = λ . On the other hand, λ = c f / ω is a decreasing function of ω which is
precisely equal to the required value to fit m half wavelengths in the 0y direction: λm( ,0) c
2
= π c f / ωm( c,0) =
2b m
[5.128]
Grazing incidence θ m = 0 corresponds to the plane guided waves m,n = 0 which are purely longitudinal in the Ox direction. Finally, wave energy is propagated at the group velocity [5.119] written here as: cm( gr,0) =
dω = c f cos θ m dk
[5.129]
As expected, no energy is propagated at normal incidence and at grazing incidence (plane guided waves) energy is propagated at the speed of sound in infinite fluid c f . 5.2.3
Cylindrical waveguides
Whatever the shape of the waveguide may be, the propagation properties are qualitatively the same as those just described based on the rectangular geometry. The major difference lays in the exact value of the cut-off frequencies, which in all cases are proportional to the speed of sound c f in the unbounded medium and
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397
inversely proportional to the largest transverse length of the waveguide. As an exercise, the case of circular cylindrical tube is considered here. The problem is formulated as follows: ∂ 2p 1∂ p 1 ∂ 2p ∂ 2p ⎛ ω + + + +⎜ ∂ r 2 r ∂ r r 2 ∂ θ 2 ∂ z 2 ⎜⎝ c f ∂p ∂r
R
2
⎞ ⎟⎟ p = 0 ⎠
[5.130]
=0
The guided mode shapes are written as: ⎛ α r ⎞ ⎧cos mθ ⎫ pg ϕ m( ,n ) ( x, y , z ) = J m ⎜ m ,n ⎟ ⎨ ⎬ pm , n ( z ) ⎝ R ⎠ ⎩ sin mθ ⎭ m = 0,1, 2,.... ; n = 0,1, 2,....
[5.131]
where α m ,n are the roots of J m′ (α m ,n ) . The longitudinal wave profile pm ,n ( z ) is governed by the differential equation: ⎡⎛ ω pm ,n ( z ) ⎢ ⎜ ⎢⎜⎝ c f ⎣
2
⎞ ⎛ α m ,n ⎞ ⎟⎟ − ⎜ ⎟ ⎠ ⎝ R ⎠
2
⎤ d2p m ,n ⎥+ =0 dz 2 ⎥ ⎦
[5.132]
To establish equation [5.132] we use the fact that J m (α m ,n r / R ) verifies the Bessel equation, hence: 2
2
α d 2 J m 1 dJ m ⎛ m ⎞ + − ⎜ ⎟ J m = − m2,n J m 2 dr r dr ⎝ r ⎠ R
[5.133]
Whence the guided wave modes: pm ,n ( z ) = am ,n e
− km ,n z
where km ,n
⎛ω =i ⎜ ⎜c ⎝ f
2
⎞ ⎛ α m ,n ⎞ ⎟⎟ − ⎜ ⎟ ⎠ ⎝ R ⎠
2
[5.134]
Since α 0,1 = 0 , the plane wave mode corresponds to the indices m = 0, n = 1, and the cut-off frequency below which only plane waves are transmitted is: c ω0,2 =
c f α 0,2 R
where α 0,2 1.84
[5.135]
Considering for instance a tube R = 1 m containing water, the plane wave model is valid up to frequencies of about 440 Hz. At 100 Hz, the magnitude of a guided
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Fluid-structure interaction
wave (0,2) is reduced by a factor ten as it travels a distance Lz 1, 25R from the source. 5.3. Forced waves Forced wave equations
5.3.1
Referring to equation [1.54] of Chapter 1, the forced motion of an inviscid fluid in the linear elastic domain is governed by the set of the three following equations: p = ρ c 2f e ∂ρ ∂ m( ) + ρ f div X f = ∂t ∂t ( e ) ρ f X f + grad p = f ( r ; t )
[5.136]
We recall that the source term arising in the equation of mass is the rate of mass per unit volume of external fluid which is injected at time t and current position r e into the fluid. Using the I.S. of units, m ( ) is expressed in kgm -3 and the source term is a mass flux density, expressed in kgm -3 s −1 . The physical meaning of such a source term is made particularily clear when the external material is injected into a control volume (Vf ) through the boundary surface (S f ) , as is the case in all applications in the present book. Thus, the rate of fluid injected into (Vf
)
may be
written as: ⌠ ⎮ ⎮ ⎮ ⎮ ⌡(Vf
)
e ⌠ ⌠ e ∂ m( ) e dV = ⎮⎮ ρ f V ( ) .n dS = ⎮⎮ ρ f q( ) dS ∂t ⌡(S f ) ⌡(S f )
[5.137]
e V ( ) is the velocity field prescribed to the injected fluid at each point of (S f ) and e e q ( ) = V ( ) .n is the volume velocity across (S f ) per unit area, quantity expressed in
( ms )
-1
. Finally, it is worth emphasising that the the one-dimensinal version of such
a volume velocity source concentrated at a surface was already introduced in Chapter 4, subsection 4.3.1.1. On the other hand, the source term arising in the momentum equation is a force density per unit volume. Its physical meaning is made particularily clear when the fluid is forced by an external pressure field exerted on the boundary surface (S f ) of
the control volume (Vf ) , as is the case in most applications in the present book. The
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399
one-dimensinal version of such a pressure source was already introduced in Chapter 4, subsection 4.3.1.2. If pressure is eliminated between the three equations [5.136], the following forced wave equation is obtained: e e ρ f X f − ρ f c 2f ΔX f = f ( ) ( r ; t ) − grad m ( ) [5.138] e Equation [5.138] is of the canonical form M f ⎡⎢ X f ⎤⎥ + K f ⎣⎡ X f ⎦⎤ = F ( ) ( r ; t ) as it ⎣ ⎦ should be. If the displacement field is eliminated, the following forced wave equation is obtained:
Δp −
5.3.2
( e) ∂ 2 m( e) 1 ∂2 p = div f − c 2f ∂ t 2 ∂t2
[5.139]
Forced waves in rectangular enclosures
For solving forced wave problems governed by equations [5.138], or [5.139], it is of particular importance, from both the theoretical and the practical points of view, to focus first on the particular case of concentrated source terms. In the context of the one-dimensional problems already analysed in Chapter 4, two distinct concentrated sources were brought in evidence, namely the volume velocity and the pressure sources. In the three-dimensional case, based on the right-hand side of the forced wave equation, one is naturally led to distinguish between the injected mass and the force sources, which in fact may also be interpreted as volume velocity and pressure sources. Furthermore, one is also led to distinguish between point, line and surface sources. Finally, as in the case of solids, all of them can be described analytically by using the appropriate Dirac distributions, namely a Dirac delta and a dipole, whence the terminology of monopole and dipole sources. On the other hand, it is extremely useful to particulatize the problem further by restricting first the study to the determination of the so-called Green’s functions, that is the acoustical responses to a unit impulsive and point source, as already mentioned in Chapter 4, at the end of subsection 4.3.2.2. It is recalled that the Laplace or Fourier transform of a Green function is a transfer function. We have already used extensively such entities to analyse discrete and continuous solid systems. Due to the relative simplicity of the forced wave equation [5.139] analytical determination of the Green functions is possible for a few simples geometries. 5.3.2.1 Green function In equation [5.139], an impulsive volume velocity source concentrated at a point r0 = x0i + y0 j + z0 k is characterized by the singular distribution:
400
−
Fluid-structure interaction
e e ∂ 2 m( ) ( r ; t ) ∂M ( ) = − δ ( t − t0 ) ∩ δ ( x − x0 ) ∩ δ ( y − y0 ) ∩ δ ( z − z0 ) ∂t2 ∂t
[5.140]
where M ( ) is the total mass of external fluid injected. As already introduced in [AXI 05], the symbol ∩ means that the application point of the impulsion occurring at t = t0 is located at the intersection of the lines x = x0 , y = y0 and z = z0 . Pertinence of the expression [5.140] can be easily checked by carrying out the necessary integrations to obtain the action of the singular source term: e
+∞
⌠ ⎮ ⎮ ⎮ ⎮ ⎮ ⎮ ⎮ ⌡−∞
⌠ ⎮ ⎮ ⎮ ⎮ ⌡(Vf
− )
where Q (
e e ∂ 2 m( ) ( r ; t ) ∂M ( ) d dt = − = − ρ f Q (e) V ∂t 2 ∂t e)
[5.141]
is the magnitude of the volume velocity source expressed in m 3s-1 .
Considering now an impulsive point source of unit magnitude, the Green function, denoted G, is the solution of the following forced problem: ∂ 2G ∂ 2G ∂ 2G 1 ∂ 2 G + + − = − ρ f δ ( t − t0 ) ∩ δ ( x − x0 ) ∩ δ ( y − y0 ) ∩ δ ( z − z0 ) ∂ x 2 ∂ y 2 ∂z 2 c 2f ∂ t 2 + ( C.B.C. ) and G ≡ 0 ; G ≡ 0 if t ≤ t0
[5.142] where once more (C.B.C.) stands for conservative boundary conditions. The solution, written as G ( r , r0 ; t − t0 ) , is the pressure response at position r = xi + yj + zk to a unit volume velocity impulsive point source (also called unit monopole point source) located at r0 = x0i + y0 j + z0 k and occuring at time t0 . The Laplace transform of [5.142] is written as: ∂ 2G ∂ 2G ∂ 2G s 2G + + − 2 = − ρ f e − st0 ∩ δ ( x − x0 ) ∩ δ ( y − y0 ) ∩ δ ( z − z0 ) ∂ x 2 ∂ y 2 ∂z 2 cf
[5.143]
+ ( C.B.C. )
Here s = iω denotes the Laplace variable and G ( r , r0 ; s ) = TL ⎡⎣G ( r , r0 ; t ) ⎤⎦ is in fact
a transfer function. Solution of the problem [5.142] and then [5.143] can be expanded as a modal series keeping in line with the results presented in Chapter 4, subsection 4.3.2.2. Details of the calculation process need not to be repeated here. The result is the triple series:
3D Sound waves
p p ∞ ∞ ∞ ρ c2 ϕ ( ) ( r ) ϕ (,m),n ( r0 ) G ( r , r0 ; s ) = f f e − st0 ∑∑∑ ,m ,n 2 2 Lx Ly Lz m n μ , m , n ( ω , m , n + s ) ρ c2 G ( r , r0 ; t − t0 ) = f f Lx Ly Lz
p sin (ω ,m ,n ( t − t0 ) ) p ϕ (,m),n ( r ) ϕ (,m),n ( r0 ) ∑∑∑ μ , m , n ω , m , n m n ∞
∞
401
[5.144]
∞
In the same way as for its 1D counterpart, formula [5.144] shows that the Green function remains unchanged if the position of the source and that of the receptor are permuted, as should be. The mode shapes are of the kind: ⎛ mπ y ⎞ ⎛ nπ z ⎞ ⎟⎟ cos ⎜ ⎟ ⎝ Lz ⎠ ⎝ Ly ⎠ n = 0,1, 2,...
ϕ (,m),n ( x, y , z ) = ϕ ( p
m = 0,1, 2,... ;
p)
( x ) cos ⎜⎜
The analytical expression of the longitudinal mode shape ϕ (
[5.145]
p)
( x)
can be specified,
once the longitudinal boundary conditions are known. The coefficient μ ,m ,n is related to the norm and the shape of the modes: μ ,m ,n =
1 Lx Ly Lz
⌠ ⎮ ⎮ ⎮ ⌡(Vf )
(ϕ ( )
p ,m ,n
( r ))
2
dr
[5.146]
For obtaining the pressure response to an impulsive monopole source of magnitude e e Q ( ) concentrated at a point, it suffices to multiply the relations [5.144] by Q ( ) . As already emphasized in subsection 5.1.2.1, the practical interest of formulas [5.144] is restricted to the low frequency range since the modal density of 3D enclosures is usually far too high for actually carrying out the triple summation involved in the series. 5.3.2.2 Response to a velocity source distributed over a surface Let us consider now the case of considerable interest in practice of a volume velocity source distributed over a wall of the rectangular enclosure, for instance that at x = 0. Typically, this kind of source is produced by the transverse vibration of the wall, which acts as an external source of volume velocity, provided its motion of complex amplitude X ( y , z; t ) i is prescribed independently of the acoustical response of the enclosure, see Figure 5.18. The problem is governed by the following equation: ∂2 p ∂2 p ∂2 p 1 ∂2 p + + − = − ρ f X ( y, z; t ) δ ( x ) ∂ x 2 ∂ y 2 ∂z 2 c 2f ∂ t 2 + ( C.B.C.
)
and p ≡ 0 ;
p ≡ 0 if t ≤ t0
[5.147]
402
Fluid-structure interaction
The Laplace transformation of [5.147] follows as: ∂ 2 p ∂ 2 p ∂ 2 p s 2 + + − p = − ρ f s 2 X ( y , z; s ) δ ( x ) ∂ x 2 ∂ y 2 ∂z 2 c 2f
[5.148]
+ ( C.B.C. )
Projection onto the modal basis gives the following solution: p ρ c2 s2 ϕ ( ) (r )Q p ( r ; s ) = f f ∑ ∑ ∑ ,m ,n 2 ,m ,n2 Lx Ly Lz m n μ,m ,n (ω,m ,n + s ) Q,m ,n =
L
⌠ z ⎮ ⌡0
L
⌠ y ⎮ ⎮ ⌡0
[5.149]
X ( y, z; s ) ϕ ,m ,n ( 0, y , z ) dydz ( p)
Figure 5.18. Sound waves forced in a rectangular duct by the vibration of a transverse wall
A first particularly simple example is that of a piston like motion, that is X is assumed to be uniform X = X 0 ( t ) . Using the mode shapes [5.145], it is immediately checked that only the plane wave modes are excited: Q,m ,n = X 0 ( s ) ϕ (
p)
⎧ Ly Lz if m = n = 0 ⎩ 0 otherwise
( 0) ⎨
[5.150]
As already pointed out in Chapter 4, if there is a pressure node at x = 0, no wave is excited by the volume velocity source. In the case of pressure antinodes, the mode shapes are: ϕ (
p)
⎛ π x ⎞ ⎟ ; = 0,1, 2,... ⎝ Lx ⎠
( x ) = cos ⎜
[5.151]
3D Sound waves
403
The response [5.149] takes on the particular form: ∞ ρ c2 X ( s ) ⎛ cos ( π x / Lx ) ⎞ ⎜1 + 2s 2 ∑ ⎟ p ( r ; s ) = f f 0 2 2 ⎜ Lx =1 ( ω ,0,0 + s ) ⎟ ⎝ ⎠ t ⌠ ⎛ ⎞ ∞ ⎮ ⎛ π c f ( t − τ ) ⎞ ⎟ X t ⎜ ( ) 2 2 0 ⎮ +∑ cos ( π x / Lx ) ⎮ X 0 (τ ) sin ⎜ p (r;t ) = ρ f cf ⎜ ⎟ dτ ⎟ Lx =1 π c0 ⎮ ⎝ ⎠ ⎟ ⎜ Lx ⎮ ⌡0 ⎝ ⎠
[5.152] The first term of the series stands for the quasi-static pressure induced by the displacement of the piston. The other terms, proportional to the acceleration of the piston, stand for the dynamical pressure response. Of course, if a pressure node is assumed to take place at x = Lx , the pertinent pressure mode shapes are: ϕ (
p)
⎛ ( 2 + 1) π x ⎞ ⎟ = 0,1, 2,... 2 Lx ⎝ ⎠
( x ) = cos ⎜
[5.153]
Accordingly, only the dynamical part of the response remains, as it should, since the enclosure is open! Finally, The one-dimensional response [4.173] can be easily e recovered simply by substituting s = iω and −ω 2 Ly Lz X 0 ( iω ) = iω Q ( ) into the Laplace transform [5.152], as could be anticipated. The second simple example worthy of a short description is that of a uniform motion restricted to a rectangular strip of the wall: h h ⎧ ⎪1 if z 0 − ≤ z ≤ z 0 + X ( y , z; t ) i = X 0 ( t )Ψ ( y , z ) i ; Ψ ( y , z ) = ⎨ 2 2 ⎪⎩ 0 otherwise
[5.154]
The following relations concerning the generalized excitation are immediately obtained, with respect to the modal basis [5.145]:
Ψ m ,n
⎧ 0 if m ≠ 0 ⎪ ⎪ ⎪⎪ Ly h if m = n = 0 =⎨ ⎪ ⎪ 2L L ⎛ nπ h ⎞ ⎛ nπ z0 ⎞ ⎪ y z sin ⎜ ⎟ cos ⎜ ⎟ if m = 0 ; n ≠ 0 ⎪⎩ nπ ⎝ 2 Lz ⎠ ⎝ 2 Lz ⎠
[5.155]
404
Fluid-structure interaction
The series giving the pressure response can be separated into four distinct components, according whether the indices and n are zero or not: ∞ ∞ ∞ ∞ p ( r ; s ) = p 0,0 ( r ; s ) + ∑ p ,0 ( r ; s ) + ∑ p 0,n ( r ; s ) + ∑∑ p ,n ( r ; s ) =1
n =1
[5.156]
=1 n =1
Again the first term stands for the quasi-static pressure response: ρ c2 X ( s ) ⎛ h ⎞ p 0,0 ( r ; s ) = f f 0 ⎜ ⎟ Lx ⎝ Lz ⎠
[5.157]
The second term stands for the dynamical part of the plane wave component of the response, written as: 2 ρ f c 2f s 2 X 0 ( s ) ⎛ h ⎞ ∞ cos ( π x / Lx ) p r ; s = ) ⎜ ⎟∑ ∑ ,0 ( 2 2 Lx =1 ⎝ Lz ⎠ =1 (ω ,0,0 + s ) ∞
[5.158]
Of course, concerning the plane wave response the only difference with respect to the uniform case is the area weighting ratio h / Lz . The third term stands for the axially uniform part of the non-plane response component. It is written as: ∑ p ( r ; s ) = 2ρ ∞
n =1
0, n
f
⎛ X ( s ) ⎞ ⎛ h ⎞ ∞ 2 Lz ⎛ nπ h ⎞ ⎛ nπ z0 ⎞ cos ( nπ z / Lz ) c 2f s 2 ⎜ 0 sin ⎜ ⎟⎜ ⎟∑ ⎟ cos ⎜ ⎟ 2 2 ⎝ 2 Lz ⎠ (ω0,0,n + s ) ⎝ Lx ⎠ ⎝ Lz ⎠ n =1 nπ h ⎝ 2 Lz ⎠
[5.159] The fourth and last term is the axially non-uniform part of the non-plane response component. It is written as: ⎛ X ( s ) ⎞ ⎛ h ⎞ c 2f s 2 ⎜ 0 ⎟⎜ ⎟× n =1 ⎝ Lx ⎠ ⎝ Lz ⎠ ∞ ⎛ nπ h ⎞ ⎛ nπ z0 ⎞ cos ( π x / Lx ) cos ( nπ z / Lz ) 2 Lz sin ⎜ ∑ ⎟ cos ⎜ ⎟ n =1 nπ h (ω2,0,n + s 2 ) ⎝ 2 Lz ⎠ ⎝ 2 Lz ⎠
∑ ∑ p ( r ; s ) = ρ ∞
=1 ∞
∑ =1
∞
,n
f
[5.160]
To check qualitatively the relative magnitude of the non-plane on the plane wave component, suffices to consider a harmonic excitation, slowly swept in frequency. By virtue of the resonant nature of the responses [5.158] to [5.160], it is easy to understand that the pressure field depends highly on the frequency and mode density. On the other hand, the non-plane wave component of the response depends on the z0 location of the source. Finally, in the weighting term h we recognize a cardinal sinus which tends to a Dirac distribution as h tends to zero. Replacing this sequence by one, gives the response to a uniform line source located at z = z0 .
3D Sound waves
405
5.3.2.3 Response to a concentrated pressure, or dipole source In the momentum equation [5.136], an impulsive force source concentrated at r0 = x0i + y0 j + z0 k is characterized by the singular distribution: e e f ( ) ( r ; t ) = MV ( )δ ( t − t0 ) ∩ δ ( x − x0 ) ∩ δ ( y − y0 ) ∩ δ ( z − z0 ) [5.161] e MV ( ) stands for an external momentum exerted at time t0 on the point r0 . Accordingly, the source term in equation [5.139] is expressed in terms of Dirac dipoles as: e e div f ( ) = MVx( )δ ( t − t0 ) ∩ δ ′ ( x − x0 ) ∩ δ ( y − y0 ) ∩ δ ( z − z0 ) + MV y( )δ ( t − t0 ) ∩ δ ( x − x0 ) ∩ δ ′ ( y − y0 ) ∩ δ ( z − z0 ) + e
[5.162]
(e)
MVz δ ( t − t0 ) ∩ +δ ( x − x0 ) ∩ δ ( y − y0 ) ∩ δ ′ ( z − z0 )
As the three components of [5.162] are identical, except eventually the magnitude of the momentum discontinuity there is no inconvenience in restricting the calculation to only one of them, the first one for instance. In fact, this is equivalent to assuming an impulsive pressure force exerted across a surface of unit area centred at r0 and perpendicular to i . The forced problem for a concentrated unit dipole acting in the longitudinal direction i is thus governed by the following equation:
Pi ∂ 2G p( )
∂ x2
+
Pi ∂ 2G p( )
∂ y2
+
Pi ∂ 2G p( )
∂z 2
2 ( Pi ) 1 ∂ Gp − 2 = c f ∂t2
δ ( t − t0 ) ∩ δ ′ ( x − x0 ) ∩ δ ( y − y0 ) ∩ δ ( z − z0 ) + ( C.B.C. ) and G
( Pi ) p
G
≡0 ;
( Pi ) p
[5.163]
≡ 0 if t ≤ t0
The symbol G marks that the solution of [5.163] is conceptually related to that of the problem [5.142], and hence to the Green function, through the duality principle between the pressure and volume velocity variables. The superscript ( Pi ) indicates
( Pi )
the direction of the dipole. It is a straightforward task to show that H p G
( Pi ) p
and
can be expanded as:
( Pi )
Hp
( Pi )
= G p
( r , r0 ; s ) =
c 2f Lx Ly Lz
Pi G p( ) ( r , r0 ; t − t0 ) =
e − st0 ∑
c 2f Lx Ly Lz
∑ ∑μ m
∑∑
m
p ϕ (,m),n ( r )
∂x + s2 ) [5.164] ( p) ∂ϕ ,m ,n ( r0 ) sin (ω,m ,n ( t − t0 ) ) ( p) ∑n ϕ,m,n ( r ) ∂x μ , m , n ω , m , n n
,m ,n
(ω
p ∂ϕ (,m),n ( r0 )
2 ,m ,n
406
Fluid-structure interaction
As an example, let us consider an external pressure field P (
e)
(t )
exerted
uniformly over the cross-section at x = 0. Using the mode shapes [5.145], it is immediately found that only the plane wave modes are excited: Q,m ,n = P (
e)
⎧ Ly Lz if m = n = 0 ⎩ 0 otherwise
( s ) ϕ ′( p ) ( 0 ) ⎨
[5.165]
As already pointed out in Chapter 4, if there is a pressure antinode at x = 0, no wave is excited by a pressure source. In the case of free axial boundary conditions (pressure nodes) the mode shapes are: ϕ (
p)
⎛ π x ⎞ ⎟ ⎝ Lx ⎠
( x ) = sin ⎜
= 1, 2,...
[5.166]
The pressure response is then found to be: 2P( p ( r ; s ) =
e)
(s) cf
Lx
⎛ π c f ⎞ sin ( π x / Lx ) ⎟ 2 2 =1 ⎝ Lx ⎠ ( ω ,0,0 + s ) ∞
∑⎜
t
p (r;t ) =
2c f Lx
∞
∑ sin ( =1
⌠ ⎮ e π x / Lx ⎮⎮ P ( ) ⎮ ⎮ ⌡0
)
⎛ π c ( t − τ ) ⎞ (τ ) sin ⎜ f ⎟ dτ Lx ⎝ ⎠
[5.167]
It is not difficult to verify that the result [5.167] does agree with the onedimensional formula [4.174] as it should. 5.3.2.4 Modal expansion method for coupled enclosures In the same way as complicated structures may often be described as an assembly of more simple substructures, attached to each other at various places by connecting elements of negligible size, complicated acoustic circuits and enclosures can often be described as an assembly of more simple piping and enclosures subsystems, communicating through small orifices. The modal substructuration method described in [AXI 05] in the context of structural elements can also be used in the context of room and piping system acoustics. It is of interest to study, how the general procedure available to deal with a formulation in terms of displacement field can be adapted to the case of a formulation in terms of pressure. It suffices to consider the assembly of two rectangular enclosures set in communication by an orifice, see Figure 5.19. The object is to determine the acoustic modes of the assembled system in terms of the modes of each subsystem. As in the case of structures, the connection between the two enclosures is modelled as interaction forces concentrated at a connecting point. Hence, in a first step at least, size of the orifice is assumed to be negligible in respect to the acoustic wavelengths of interest.
3D Sound waves
407
Figure 5.19. Two rectangular enclosures connected to each other by a small orifice
Furthermore, based on physical reasoning, it turns out that the orifice can be suitably modelled as a small tube, cross sectional area S0 of which is equal to that of the real orifice and length L0 is smaller than the wavelengths of interest (cf. Chapter 2, subsection 2.2.2.4). Fluid interaction forces can be modelled as a coumpound of inertia and elastic forces. Using the notations specified in Figure 5.19, the coupled modal system written in terms of the displacement fields, reads as: 1 1 1 2 1 2 ρ f c 2f ΔX (f ) + ω 2 ρ f X (f ) = K L X (f ) − X (f ) − ω 2 M L X (f ) + X (f ) δ ( r − r0 )
( (
)
))
(
(1)
+ ( C.B.C. ) f [5.168] 2 2 1 2 1 2 ρ f c 2f ΔX (f ) + ω 2 ρ f X (f ) = − K L X (f ) − X (f ) − ω 2 M L X (f ) + X (f ) δ ( r − r0 )
( (
)
(
))
(2)
+ ( C.B.C. ) f
K L and M L are the stiffness and mass coefficients per unit area of the connecting element, respectively. They can be considered either as numerical parameters suitably adjusted in accordance with the truncation order of the modal bases, as already explained in [AXI 05] Chapter 4, or as physical parameters related to the actual properties of the connecting element:
408
Fluid-structure interaction
K L = ρ f c 2f / L0
[5.169]
M L = ρ f L0
Qualitatively, if K L tends to infinity, the relative motions of the fluid at the ends of the virtual connecting tube tend to vanish. Hence, K L can be suitably used as a penalty parameter to ensure the condition of fluid incompressibility with a high degree of accuracy. On the contrary, if M L tends to infinity, the motion of the fluid inside the tube tends to vanish. Therefore, M L can be used either to describe a realistic inertia effect through the connecting element, or as a penalty parameter, to block the fluid motion. Finally, as in the case of solid structures, the displacement modes appropriate to carry out a meaningful projection correspond to an “open” (i.e. pressure node) condition at the connecting orifice, in such a way that, at least, some generalized interaction forces are nonzero. The remaining mathematical procedure follows exactly along the same lines as in the case of solids. The modal expansion of the non trivial solutions of [5.168] are written as:
(
( X ) ( X ) ( X ) ϕ ,m ,n = ∑ αi , j ,kϕ i , j ,1k ( r ) + βi , j ,kϕ i , j ,2k ( r ) i , j ,k
)
[5.170]
( X ) The acoustic mode shapes in terms of the displacement field are denoted ϕ ,m ,n for ( X ) ( X ) the connected enclosures, ϕ i , j ,1k ( r ) for the first enclosure and ϕ i , j ,2k ( r ) for the
second enclosure. Projection of the modal equation [5.168] onto the modal basis ( X ) ( X ) ϕ i , j ,1k ( r ) , ϕ i , j ,2k ( r ) i, j, k = 0,1, 2,... gives the algebraic equation:
{
ρf
}
((ω
)
2 (1) ,m ,n
)
− ω 2 μ(1),m ,nα ,m ,n =
⎛ ∞ ⎛ ∞ ( X ) ⎞ ( X ) ⎞ − ( K L − ω 2 M L ) ⎜ ∑ αi , j ,kϕi , j ,1k ( r0 ) ⎟ − ( K L + ω 2 M L ) ⎜ ∑ βi , j ,kϕi , j ,2k ( r0 ) ⎟ = 0 ⎝ i , j ,k = 0 ⎠ ⎝ i , j ,k = 0 ⎠ ρf
((ω )
2 (2) ,m ,n
)
[5.171]
− ω 2 μ(2) , m , n β ,m ,n =
⎛ ∞ ⎛ ∞ ( X ) ⎞ ( X ) ⎞ − ( K L − ω 2 M L ) ⎜ ∑ βi , j ,kϕi , j ,2k ( r0 ) ⎟ − ( K L + ω 2 M L ) ⎜ ∑ βi , j ,kϕi , j ,2k ( r0 ) ⎟ = 0 ⎝ i , j ,k = 0 ⎠ ⎝ i , j ,k = 0 ⎠
where the coefficients μ(,m) ,n and μ(,m) ,n are related to the norm of the mode shapes 1
2
in the same way as in formula [5.146]. Provided the modes of each individual enclosure are suitably ordered, typically, according to the rule of increasing frequency, equation [5.171] can be rewritten as a standard modal equation which holds for any conservative and discrete system:
3D Sound waves
409
c c ⎡ ⎡ ⎡ ⎡ K (1) ⎤ + ⎡ K ( c ) ⎤ ⎤ ⎡ ⎡ ⎡ M (1) ⎤ + ⎡ M ( c ) ⎤ ⎤ ⎤⎤ − ⎡⎡ K ( ) ⎤⎤ ⎤ + ⎡ ⎡ M ( ) ⎤⎤ ⎦ ⎣ ⎦⎦ ⎦⎦ ⎥ ⎦ ⎣ ⎦⎦ ⎦⎦ ⎣⎣ ⎣⎣ ⎢⎢⎣⎣ ⎥ ⎥ ⎡ [α ] ⎤ = ⎡[ 0]⎤ 2 ⎢⎣⎣ ω − ⎢⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ c c ⎡ ⎡ K (2) ⎤ + ⎡ K (c) ⎤ ⎤ ⎥ ⎡ ⎡ M ( 2 ) ⎤ + ⎡ M ( c ) ⎤ ⎤ ⎥ ⎥ ⎣[ β ]⎦ ⎣[ 0]⎦ − ⎡K ( ) ⎤ + ⎡M ( ) ⎤ ⎢ ⎣⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎦ ⎣ ⎦⎦ ⎣
[5.172] 1 2 1 2 where ⎡⎣ K ( ) ⎤⎦ , ⎡⎣ K ( ) ⎤⎦ and ⎡⎣ M ( ) ⎤⎦ ⎡⎣ M ( ) ⎤⎦ are the diagonal stiffness and mass c c matrices of the individual enclosures. ⎡⎣ K ( ) ⎤⎦ and ⎡⎣ M ( ) ⎤⎦ are the full stiffness and mass matrices which couple the generalized displacement [α ] and [ β ] . The
formalism is illustrated by considering the simple 1D problem of the plane wave modes of two identical enclosures, or tubes, defined by the following numerical parameters: ρ1 = ρ 2 = ρ 0 = 1.2 kgm -3 ; c1 = c2 = c0 = 340 m s -1 ; L1 = L2 = L0 = 1.7 m
The natural frequencies and mode shapes of the individual enclosures, open at both ends, are: ϕ n(
X1 )
⎛ nπ x ⎞ = cos ⎜ ⎟ ⎝ L0 ⎠
0 ≤ x ≤ L0 ; ϕ n(
f n( ) = f n( ) = 100n Hz 1
2
X2 )
⎛ nπ x ⎞ = cos ⎜ ⎟ ⎝ L0 ⎠
L0 ≤ x ≤ 2 L0
n = 0,1, 2....
Figure 5.20. Mode coupling by connecting two identical tubes through a lumped element of large stiffness and variable mass coefficient μc = M L / 2 ρ f L0
410
Fluid-structure interaction
The tubes are connected to each other by a lumped element characterized by a large stiffness coefficient K L >> ( ρ f c 2f / L ) N 2 , where N is the highest rank of the modes included in the model, and a mass coefficient varied as a free parameter within the range 0 ≤ M L ≤ 20ρ f L0 , which suffices to cover the whole range of interest. Using formula [5.171], we obtain: 2 ⎧ ⎛ nπ ⎞ 2 ⎪ρ c μ if j = k = n j +k K (1) ( j, k ) = ⎨ f f n ⎜⎝ L0 ⎟⎠ ; K (C ) ( j , k ) = K L ( −1) ⎪ 0 if j ≠ k ⎩
⎧ ρ μ if j = k = n j+k 1 C M ( ) ( j, k ) = ⎨ f n ; M ( ) ( j, k ) = M L ( −1) 0 if j ≠ k ⎩ ⎧ L if n = 0 μn = ⎨ 0 ⎩ L0 /2 if n ≥ 1
A few numerical results are shown in Figure 5.20, which concern the third mode of the coupled tubes. If μc = M L / 2 ρ f L0 is negligibly small, the natural frequency and the shape of the coupled mode correspond to those of the third mode of the equivalent tube of length 2L, as it should. If the value of μ c is increased, the natural frequency and the fluid motion at the connecting point decrease, as expected. Of course, the second mode of the coupled tubes also corresponds to the second mode of the equivalent tube of length 2L; however as there is a displacement node at the connecting point, this mode is insensitive to any change in μ c . On the other hand, it changes with the value of the stiffness parameter γ c = K L / 2 ρ f c 2f L0 , as shown in Figure 5.21. Using now the formulation in terms of pressure, it can be noticed that the lefthand side of equation [5.138] is of the same form as the left-hand side of equation [5.139] provided X f is replaced by the dual variable − p and equation [5.138] is devided by E f = ρ f c 2f . Therefore the coupled modal system [5.168] is formulated as: Δp ( ) + 1
ω2L ω 2 (1) 1 ( 2 ) 1 2 1 p = p − p ( ) δ ( r − r0 ) + 2 0 p ( ) + p ( ) δ ( r − r0 ) 2 cf L0 cf
(
)
(
)
(1)
+ ( C.B.C. ) f Δp
(2)
ω2L ω 2 2 −1 ( 2 ) 1 2 1 + 2 p( ) = p − p ( ) δ ( r − r0 ) − 2 0 p ( ) + p ( ) δ ( r − r0 ) cf L0 cf
(
(2)
+ ( C.B.C. ) f
)
(
)
[5.173]
3D Sound waves
411
Figure 5.21. Mode coupling by connecting two identical tubes through a lumped element of 2 zero mass and variable stiffness coefficient γ c = K L / 2 ρ f c f L0
Hence, equations [5.168] and [5.173] are mathematically equivalent to each other and any result valid for one of them also holds for the other, provided the dual variable is considered. For instance in equations [5.173], if L0 tends to zero, fluid motion vanishes and if c f tends to zero, pressure vanishes at the connection, which is the dual behaviour of the connecting terms in equations [5.168]. On the other hand, the pressure modes of the problem of two identical tubes closed at both ends are the same as the displacement modes of the dual problem of an assembly of two identical tubes open at both ends, which were presented just above as an illustrative example. 5.3.3
Forced waves in waveguides
In the same way as the forced waves in an enclosure can be expressed in terms of a modal series, the waves forced in a waveguide can be expanded as a guided wave series. The major difference between the two cases lies in the nature of the modal waves involved. Indeed, so long as an enclosure is modelled as a conservative system, the acoustical modes are standing waves and so the forced waves. As a consequence, no mechanical energy is radiated. In contrast, guided wave modes are travelling waves and so the forced waves.
412
Fluid-structure interaction
5.3.3.1 Local and far acoustical fields As shown in Figure 5.22, we consider a rectangular waveguide of axis Ox excited by the axial motion prescribed to a membrane stretched at the inlet crosssection. As demonstrated in subsection 5.3.2.2, in the case of a finite box, this standard problem can be solved, formally at least, by applying the usual machinery of the modal projection method. The extension of the method to the case of a waveguide is straightforward. Assuming the walls of the waveguide are fixed, the forced problem is written as: ∂2p ∂2p ∂2p ⎛ ω + + +⎜ ∂ x 2 ∂ y 2 ∂ z 2 ⎜⎝ c f ∂p ∂y
y =− b
∂p = ∂y
y =b
∂p = ∂z
2
⎞ 2 ⎟⎟ p = ρ f ω X ( y , z )δ ( x ) ⎠
z =− a
∂p = ∂z
[5.174]
=0 z =a
Figure 5.22. Rectangular waveguide excited by the prescribed harmonic displacement of a membrane stretched at the inlet
We seek a solution in terms of the forward guided modes written as: p(
+)
∞
∞
m =0
n =0
( x, y, z;ω ) = eiω t ∑ ∑ Am( +,n) (ω ) pm( +,n) ( x, y , z )
[5.175]
For convenience, the results established in subsection 5.2.2 which are needed here are collected as:
3D Sound waves
413
⎛ mπ ( y + b ) ⎞ ⎛ nπ ( z + a ) ⎞ i (ωt − km ,n x ) + pm( ,n) ( x, y , z ) eiωt = cos ⎜ ⎟ cos ⎜ ⎟e 2b 2a ⎝ ⎠ ⎝ ⎠ +b
μm2 ,n =
k
2 m ,n
⌠ ⎮ ⎮ ⎮ ⎮ ⎮ ⌡− b
⎛ω =⎜ ⎜ cf ⎝
+a
⌠ ⎮ ⎮ ⎮ ⎮ ⎮ ⌡− a
2
⎛ ⎛ mπ ( y + b ) ⎞ ⎛ nπ ( z + a ) ⎞ ⎞ ⎜⎜ cos ⎜ ⎟ cos ⎜ ⎟ ⎟⎟ dydz 2b 2a ⎝ ⎠ ⎝ ⎠⎠ ⎝
[5.176]
2
⎞ ⎧⎪⎛ mπ ⎞ 2 ⎛ nπ ⎞2 ⎫⎪ ⎟⎟ − ⎨⎜ ⎟ +⎜ ⎟ ⎬ m = 0,1, 2,... ; n = 0,1, 2,... ⎠ ⎩⎪⎝ 2b ⎠ ⎝ 2a ⎠ ⎭⎪
However, in contrast with the standing waves case, direct substitution of the series [5.175] into the partial derivative equation [5.174] fails because by definition the guided modes are solution of the homogeneous version of the same equation, and this whatever the value of the angular frequency may be, see equation [5.107]. Fortunately, the difficulty disappears if the system [5.174] is written in terms of ordinary functions as: ∂2p ∂2p ∂2p ⎛ ω + + +⎜ ∂ x 2 ∂ y 2 ∂ z 2 ⎜⎝ c f ∂p ∂y ∂p ∂x
= y =− b
∂p ∂y
= y =+ b
∂p ∂z
2
⎞ ⎟⎟ p = 0 ⎠ = z =− a
∂p ∂z
=0
[5.177]
z =+ a
= ρ f ω 2 X ( y, z ) x =0
Substituting the series into [5.177], the homogeneous partial derivative equation is identically verified, as should be, and the unknown coefficients of the series [5.175] can be determined by using the axial boundary condition, once it is projected onto the transverse shape of the guided mode. The result is: Am( ,n) (ω ) = +
ωρ f Qm ,n μ m2 ,n k m ,n
[5.178]
where Qm ,n is the generalized volume velocity induced by the prescribed motion of the membrane: +b
Qm ,n =
⌠ ⎮ iω ⎮⎮ ⎮ ⎮ ⌡− b
+a
⌠ ⎮ ⎮ ⎮ ⎮ ⎮ ⌡− a
⎛ mπ ( y + b ) ⎞ ⎛ nπ ( z + a ) ⎞ X ( y , z ) cos ⎜ ⎟ cos ⎜ ⎟ dydz 2b 2a ⎝ ⎠ ⎝ ⎠
[5.179]
414
Fluid-structure interaction
In agreement with the results established in subsection 5.2.2 considering a monochromatic source pulsating at ω 0 , sufficiently far from the source the forced wave contains only the guided wave components which comply with the cut-off condition [5.115]: 2
2
⎛ mπ ⎞ ⎛ nπ ⎞ (c) cf ⎜ ⎟ +⎜ ⎟ = ωm , n < ω0 b a 2 2 ⎝ ⎠ ⎝ ⎠
[5.180]
Of course, the cut-off frequency is zero for the plane wave guided mode, and it is an easy task to check that the corresponding waveguide response is identical to the tube response according to the plane wave approximation already analysed in Chapter 4. With the aid of [5.176] and [5.179], we obtain: 2 μ0,0 = 4ab = S f
⌠
+b
Q0,0 = iω ⎮⎮
⌡− b
[5.181] +a
⌠ ⎮ ⎮ ⌡− a
X ( y , z ) dydz
[5.182]
Finally, with the aid of [5.178], the plane wave response is found to be: ( ) p0,0 ( x; t ) = +
ρ f c f Q0,0 Sf
e
⎛ x⎞ iω ⎜ t − ⎟ ⎝ c0 ⎠
= Z0,0Q0,0 e
⎛ x⎞ iω ⎜ t − ⎟ ⎝ c0 ⎠
[5.183]
Z0,0 is the impedance of the waveguide for the plane wave mode, which identifies
with that of the equivalent tube as defined in Chapter 4. On the other hand, relation [5.182] points out that to excite a plane wave, the motion of the membrane must have a nonzero mean component. Differently stated, the membrane, or plate motion, must induce a change in the fluid volume, as could be anticipated for a volume velocity source. Therefore, the same forced wave could be excited by prescribing the same mean displacement to a rigid piston at the inlet of the waveguide. As a final example, let us consider the response to the anti-symmetrical source defined as: ⎧ + X 0 if 0 ≤ z ≤ a X ( y, z ) = ⎨ ⎩ − X 0 if − a ≤ z ≤ 0
[5.184]
The nonzero generalized volume velocities are found to be: Q0,2 n +1 = −iω
2 X 0 S f ( −1)
( 2n + 1) π
n
n = 0,1, 2...
[5.185]
3D Sound waves
415
As should be expected, only the modes which are anti-symmetric about the Oy axis are excited. The pressure wave is found to be: p( x, z, ω ; t ) =
−2iX 0 ρ f c 2f a
∞
∑g n =0
n
⎛ aω ⎞ ⎛ π ( 2n + 1)( z + a ) ⎞ i (ωt − k0,n x ) ⎜⎜ ⎟⎟ cos ⎜ ⎟e 2a c ⎝ ⎠ ⎝ f ⎠
⎛ aω ⎞ ( −1) where g n ⎜ = ⎟ ⎜c ⎟ ⎝ f ⎠ ⎛ ( 2n + 1) π ⎞ ⎛ c f ⎞ 2 ⎛ ⎛ ( 2n + 1) π ⎜ ⎟ ⎜ ⎟ − ⎜⎜ 2 2 ⎝ ⎠ ⎝ aω ⎠ ⎜⎝ ⎝ n
[5.186] ⎞ ⎛ c0 ⎞ ⎟⎜ ⎟ ⎠ ⎝ aω ⎠
2
⎞ ⎟⎟ ⎠
2
The longitudinal velocity field is: v x ( x, z , ω ; t ) =
i ∂p ⇒ ωρ f ∂x
⎛ π ( 2n + 1)( z + a ) ⎞ i (ωt − k0,n x ) v x ( x, z, ω ; t ) = −4iω X 0 ∑ cos ⎜ ⎟e 2a n = 0 ( 2n + 1) π ⎝ ⎠ ∞
( −1)
n
[5.187]
The results are illustrated in Figures 5.23 and 5.24, taking the example of a rectangular waveguide (a = 5 m, b = 1 m) filled with air at STP. The series was computed by retaining the first twenty odd modes of ranks (0,n).The forced wave of Figure 5.23 is evanescent in nature since its frequency is below the first cut-off frequency f 0,1c = c f / 4a . So the only nonvanishing component is the local field which takes place in the immediate vicinity of the source. It may be noticed that the profile of the velocity field at x= 0 agrees with the analytical shape [5.184], except the Gibbs oscillations induced by the modal truncature, as should be. As soon as x increases, the profile is smoothed out due to the low pass filtering effect associated with wave evanescence and beyond x 2a , it is essentially flat and nearly equal to zero. The profiles of the pressure field along the longitudinal direction Ox are similar to those of the velocity field. In the transverse z direction Oz, they are significantly smoother, especially near the source. The frequency of the forced wave of Figure 5.24 is ten per cent above the first cut-off frequency. So both the local and the far acoustic fields can be observed. Actually, as indicated in Figure 5.24, the wave travels along the 0x direction without any attenuation according to the first non-plane guided mode. Near the source, contribution of the evanescent modes are as important as in the first case.
416
Fluid-structure interaction
Figure 5.23. Evanescent wave f exc < f 0,1 = 17 Hz
3D Sound waves
417
Figure 5.24. Travelling wave f exc = 1.1 f 0,1
5.3.3.2 Impedance surface and mode coupling The analysis made in Chapter 4 subsection 4.2.1.3 concerning the reflection and transmission of plane waves at a change of impedance are extended here to the case
418
Fluid-structure interaction
of guided waves. In contrast with the plane wave model, the impedance can now vary from one point to another in the same cross-section. Accordingly, even if the incident wave comprises a single mode, the reflected and transmitted waves are a compound of several guided modes as soon as the impedance varies in the transverse directions. To demonstrate the mechanism of guided mode coupling, let us consider the example of a semi-infinite tube extending from x → −∞ and terminated at x = 0 by a surface impedance Zout ( y , z ) , see Figure 5.25. A forward guided mode of transverse ranks m and n is emitted from −∞ and is reflected by the surface impedance Zout ( y , z ) at x = 0. The incident pressure and longitudinal velocity waves are written as: pm( ,n) = ϕ m( ,n ) ( y , z ) e ( +
um( ,n) = − +
i ω t + km ,n x )
pg
km ,n ωρ f
ϕ m( ,n ) ( y , z ) e ( pg
i ω t + km ,n x )
=−
pm( +,n)
[5.188]
( sp )
Zm ,n
where the specific modal impedance is defined as: Zm( ,n ) = sp
pm( ,n) +
(+)
um ,n
=−
ρfω k m ,n
= − ρ f cm( ,n) ψ
[5.189]
Figure 5.25. Waveguide extending in the x <0 range and terminated by a surface impedance at x = 0
The acoustic wave within the tube is the resultant of the incident wave plus all the waves reflected at the surface impedance. Such a resultant wave is conveniently written as:
3D Sound waves
(
p ( ) = ϕ m( ,n ) ( y , z ) e pg
−
u(−)
ikm ,n x
+ Rm ,n e
− ikm ,n x
∞
∞
)+ ∑ ∑R m ' ≠ m n ′≠ n
ϕ m( ',n)' ( y , z ) e
m ', n '
pg
419
− ikm ',n ' x
∞ ∞ 1 ⎧ ikm ,n x x⎫ − ik x − ik ( pg ) = − Rm ,n e m ,n − ∑ ∑ k m ',n ' Rm ',n 'ϕ m( pg',n)' ( y , z ) e m ',n ' ⎬ ⎨km ,nϕ m ,n ( y , z ) e ρfω ⎩ ⎭ m ' ≠ m n ′≠ n
(
)
[5.190] where the time harmonic factor common to all terms has been dropped for alleviating the expressions. On the other hand, the negative sign used as an upper script marks the fact that the resulting wave [5.190] is defined in the x ≤ 0 range. Finally, the wave is written as the combination of a pair of forward and backward + guided modes, of rank m,n equal to that of the primary incident wave pm( ,n) , plus the contribution of all the other modes m ' ≠ m , n ' ≠ n which appear as backward travelling waves eventually induced by the reflection of the primary incident wave. The relative contributions of the individual modes are governed by the coefficients Rm ,n and Rm ',n ' , which depend on the terminal impedance. The acoustic field [5.190] must comply with the boundary condition: p( ) − u( ) −
x =0
∞ ∞ ⎛ ⎞ ρ f ω ⎜ ϕ m( pg,n ) ( y , z ) (1 + Rm ,n ) + ∑ ∑ Rm ',n 'ϕ m( pg',n)' ( y , z ) ⎟ m ' ≠ m n ′≠ n ⎝ ⎠ =Z = out ∞ ∞ ⎧ ⎫ ( pg ) ( pg ) − − k ϕ y z R k R ϕ y z , 1 , ( ) ( ) ( ) ⎨ m , n m ,n ⎬ ∑ ∑ m ',n ' m ',n ' m ',n ' m ,n ⎩ ⎭ m ' ≠ m n ′≠ n
[5.191]
Let us assume first that the terminal impedance is uniform. Since the guided mode shapes behave as linearly independent functions from one mode to the other, the only possibility left to comply with the condition of a uniform surface impedance Zout is the absence of mode coupling: Rm ',n ' ≡ 0 ∀ m ' ≠ m ; n′ ≠ n ⇒ p( ) − u( ) −
x =0
⎛ 1 + Rm ,n = ρ f cm(ψ,n) ⎜ ⎜ 1− R m ,n ⎝
⎞ ( sp ) ⎛ 1 + Rm , n ⎟⎟ = Zm ,n ⎜⎜ ⎠ ⎝ 1 − Rm ,n
⎞ ⎟⎟ = Zout ⎠
[5.192]
The results displayed in formulas [5.190] to [5.192] call for comment. First, the primary wave is reflected back as a wave of the same rank. Therefore, if Zout is uniform, the guided modes do not mix up on reflection. It is an easy task to solve [5.192] to express the coefficient of reflection Rm ,n as a function of the terminal impedance and the specific modal impedance of the guidewave Zm( ,n ) = ρ f cm( ,n) . For sp
ψ
instance, if the waveguide is terminated by a pressure node, Rm ,n = −1 which means that the m,n forward wave is fully reflected as a backward m,n wave. Reciprocally, it is immediately seen that the matched impedance which suppresses any reflection
420
Fluid-structure interaction
is precisely the specific impedance. More generally, if the impedance surface is uniform, all the results concerning the reflection and the transmission of plane waves in a tube can be readily adapted to the guided waves of any rank, by adopting the modal specific impedance [5.189], or equivalently the modal tube impedance: Zm( ,n) = +
pm( ,n)
ρ f cm( ,n) ψ
+
qm( ,n) +
=+
[5.193]
Sf
where here the positive sign in the upper script stands for a forward wave travelling from x → −∞ . Considering now the more general case of a non uniform terminal impedance, the same reasoning as above leads to the conclusion that mode coupling is needed to accommodate with the space variation of the terminal impedance. To compute the wave coefficients Rm ,n and Rm ',n ' , equation [5.191] can be restated as the linear forced equations: ∞
∞
∑ ∑ ϕ ( ) ( y, z ) ( ρ
m '=0 n '=0
pg m ', n '
ω + Zout km ',n ' ) Rm ',n ' = ϕ m( ,n ) ( y , z ) (Zout k m ,n − ρ f ω ) pg
f
[5.194]
which is projected onto the transverse mode shapes ϕ m( ,n ) to produce an algeabraic pg
linear system of the type.
( ρ ωμ f
2 m ,n
+ Z mm,,nn km ,n ) Rm ,n +
∞
∞
∑ ∑Z
m ' ≠ m n ′≠ n
m ,n m ', n ' m ', n '
k
Rm ',n ' = Z mm,,nn k m ,n − ρ f ωμm2 ,n
[5.195]
where the following coefficients are used: +b
μ
2 m ,n
=
⌠ ⎮ ⎮ ⌡− b
+a
⌠ ⎮ ⎮ ⎮ ⌡− a
(
ϕ m( ,n ) pg
)
2
⌠
dydz ; Z mm,',nn ' = ⎮⎮
+b
⌡− b
+a
⌠ ⎮ ⎮ ⌡− a
Zoutϕ m( ,n )ϕ m( ',n)' dydz pg
pg
[5.196]
Of course, if Zout is uniform the coupling coefficients Z mm,',nn ' vanish and it is easily verified that the solution of equation [5.195] fully agrees with that of [5.192]. To conclude this subsection, it may be worth mentioning that the same mechanism of mode coupling holds in the case of any non uniform impedance surface located either at an intermediate position within a waveguide or at a termination. Hence, in pipes and ducts, whenever there is a sudden accident in the tube geometry, a constriction, a change in the cross-sectional area or in the tube direction (elbows) etc., fluid motion departs, at least locally, from the plane-wave model. For more detailed information on the subject, the reader can be referred in particular to [MOR 81] and [MOR 86]. On the other hand, it is of interest, from the standpoint of the general principles of continuum mechanics at least, to point out
3D Sound waves
421
that the guided sound waves in a fluid behave in agreement with the Saint Venant principle, which is of paramount importance for modelling solid bodies as structural elements, cf. [AXI 05], Chapter 1. In the case of elastic solids it can be enuntiated as: The elastic response induced by a local force system, whose resultant force and torque are both zero, become negligible far enough from the small loaded portion of the body. In other words, if sufficiently far from the loaded domain, the response depends solely upon the resultant force and torque of the actual loading system. Restated in terms of guided acoustics, the principle could be enunciated as follows: The acoustic response induced by a local source system, whose resultant pressure and volume velocity are both zero, become negligible far enough from the loaded domain of the guidewave provided the plane wave approximation holds, that is the wave frequency is lower than the minimum cut-off frequency of the non-plane guided modes. In other words, if sufficiently far from the source, the response depends solely upon the mean pressure and volume velocity source, as averaged over the loaded cross-section. 5.3.4
Forced waves in open space: Green’s functions
5.3.4.1 3D unbounded medium Let us consider again the forced problem [5.142] written in free space as: 1 G = − ρ f δ ( r − r0 )δ (t − t0 ) 2 cf G ( r , r0 ; t − t0 ) ≡ 0 ∀ t < t0 ΔG −
[5.197]
To solve the problem in the case of a uniform and isotropic medium, it is natural to adopt the spherical coordinate system. Thus the above equation is written as: 2 dG d 2G 1 − ρ f δ ( r ) + − G= δ ( t − t0 ) r dr dr 2 c 2f 4π r 2
where r = r − r0
[5.198]
is the distance between the source point and the receptor point.
The Laplace transform of equation [5.198] follows as: 2 dG d 2G ⎛ s + −⎜ r dr dr 2 ⎜⎝ c f
2
⎞ − ρ f δ ( r ) − st0 e ⎟⎟ G = 4π r 2 ⎠
Equation [5.199] is equivalent to the system:
[5.199]
422
Fluid-structure interaction
2 dG d 2G ⎛ s + −⎜ r dr dr 2 ⎜⎝ c f
2
⎞ ⎟⎟ G = 0 ⎠
[5.200]
− ρ f − st0 dG = r2 e dr 0+ 4π
It can be verified, by substitution for instance, that the solution is of the type: ⎛
r ⎞
− s ⎜ t0 ∓ ⎟ ρ ⎜ c ⎟ G ± ( r ; s ) = f e ⎝ f ⎠ 4π r
[5.201]
Thus, reverting to the time domain: ⎛ ρ r − r0 G± ( r − r0 ; t − t0 ) = f δ ⎜ t − t0 ∓ cf 4π r ⎜⎝
⎞ ⎟⎟ ⎠
[5.202]
The Green function [5.202] is found to verify the reciprocity theorem, as expected. The subscript ± indicates the existence of two distinct travelling waves, the so called outgoing, or diverging wave which expands from the source point r0 with speed + c f as time elapses and the incoming, or converging, wave which travels from infinity with speed − c f to the point source r0 . Both of them have the expected spherical symmetry. The pressure field is concentrated at the distance r = c f ( t − t0 ) from the source point and is singular at r = ro . The velocity field associated with the pressure wave [5.202] is obtained by solving the radial component of the momentum equation: ρf
1 ⎧⎪ 1 ⎛ r ∂u ∂p ∂u =− ⇒ = ⎨ 2 δ ⎜⎜ t − t0 ∓ cf ∂t ∂r ∂ t 4π ⎪⎩ r ⎝
⎞ 1 ⎛ r δ ' ⎜ t − t0 ∓ ⎟⎟ ± ⎜ cf ⎠ rc f ⎝
⎞ ⎪⎫ ⎟⎟ ⎬ ⎠ ⎭⎪
[5.203]
Again δ ' denotes the derivative of the Dirac distribution, called the Dirac dipole. Integration of [5.203] is immediate, which yields: u=
1 4π
⎧⎪ 1 ⎛ r ⎨ 2 U ⎜⎜ t − t0 ∓ r c f ⎝ ⎩⎪
⎞ 1 ⎛ r δ ⎜ t − t0 ∓ ⎟⎟ ± ⎜ rc c f f ⎠ ⎝
⎞ ⎫⎪ ⎟⎟ ⎬ ⎠ ⎭⎪
[5.204]
The physical meaning underlying such results is clarified in the next subsections where the Green function is used as a convenient mathematical tool for solving a large variety of forced wave problems of theoretical and practical interest. Here, as a first application, we consider first a concentrated volume velocity source, or more e concisely a simple monopole source, of time varying amplitude Q ( ) ( t ) . Recalling e e that it corresponds to the source term −∂ 2 m( ) / ∂ t 2 = − ρ f Q ( ) ( t ) in the forced wave
3D Sound waves
423
equation [5.139], the response is immediately obtained by using the convolution product. Retaining the outgoing wave only, the result is: p+ ( r , t ) =
ρf 4π r
t
⌠ (e) ⎮ Q ⎮ ⌡0
e ⎛ ρ f Q ( ) (ϑ ) U (ϑ ) r ⎞ τ δ τ τ − − = t d ( ) ⎜⎜ ⎟ c f ⎟⎠ 4π r ⎝
[5.205]
where ϑ = t − r / c f is the so called retarded time. The retardation accounts for the delay necessary for the pressure response to travel the distance r from the source. Furthermore, the Heaviside step function marks the fact that p( r , t ) vanishes at any time earlier than r / c f , in agreement with the principle of causality. The velocity field related to that wave follows as: ⎛ Q ( e ) (ϑ ) Q ( e ) (ϑ ) ⎞ u+ ( r ; t ) = ⎜ + ⎟ U (ϑ ) ⎜ 4π r 2 4π rc f ⎟⎠ ⎝
[5.206]
It is of interest to discuss first the component of [5.206] which varies as r −2 , hereafter termed the near field component. Its value at time t and distance r from the point source is the same as that produced by the flow outgoing from the point source at the earlier time r / c f through the sphere of radius r . The near field component is not interpreted as an acoustic field since it exists even if Q ( ) is time independent. In that case, the acoustic pressure field must be identically zero. Actually, the near velocity field can be related to the convective inertia of the fluid driven by the source, taking also into account the travel time due to fluid compressibility, as further explained a little later on the example of a pulsating sphere. Another way to be convinced that the acoustic component of the velocity field is described by the second term of [5.206] is to calculate the sound intensity. By substituting the field variables [5.205] and [5.206] into the formula [3.26] the instantaneous acoustic intensity is found to be: e
e e ρ Q ( )Q ( ) U (ϑ ) ρ f I ( r ; t ) = p+ ( r ; t ) u + ( r ; t ) = 0 + 2 cf ( 4π ) r 3
⎛ Q ( e ) ⎜⎜ ⎝ 4π r
⎞ ⎟⎟ ⎠
2
[5.207]
The mean intensity averaged over time T is: ⎛ 1 ⌠T ⎛ 1 ⌠ T (e) ρf e) ⎞ ( ⎜ ⎮⎮ Q ( e ) I = ⎜ ⎮ Q dQ ⎟⎟ + 2 2 ⎜ T ⎮⌡0 c π r 4 ( 4π ) r 3 ⎜⎝ T ⎮⌡0 ( ) ⎠ f ⎝ ρf
( )
2
⎞ dt ⎟ ⎟ ⎠
[5.208]
Assuming either a source of period T, or a transient source of duration T, the first term in [5.208] cancels and the mean intensity can be expressed as:
424
I =
Fluid-structure interaction
(
(e) ρ f Q rms
)=
c f ( 4π r )
2
2 prms ρ f cf
[5.209]
It can be noticed that formula [5.209] is the same as that which holds in the case of plane waves. Finally, it is also of interest to notice that in the case of a harmonic wave, the acoustic component of the velocity, also termed far field, prevails over the inertial component, also termed near field, at a characteristic distance equal to the reciprocal of the wave number: ω ra / c f 1 ⇒ ra = c f / ω = k −1
[5.210]
5.3.4.2 3D medium bounded by a fixed plane, image source method An elegant and efficient way to solve radiation problems in the presence of perfectly reflecting plane boundaries is to use the so called image source method, which is introduced here by considering the case of a monopole source radiating in a uniform 3D medium limited by a plane and fixed wall, see Figure 5.26.
Figure 5.26. Field at P from a source at P0 and its image at P0′
As the reflection law [5.6] is the same as in optics, it indicates that the reflecting plane acts as a mirror and the pressure observed at P can be interpreted as resulting from a pair of spherical waves. Both are emitted at P0 ; however, one reaches P directly, while the other one is a reflected wave which seems to be emitted from the image source at the mirror point P0′ . In Figure 5.26, these waves are represented as
3D Sound waves
425
circular arcs centred at P0 (black line) and P0′ (white line) respectively. The way these waves interfere depends on the boundary conditions at the plane, or as an equivalent on the relative sign of the actual and virtual sources. At the fixed wall, the normal component of the gradient of pressure must be zero. It is not difficult to understand that to be fullfiled such a condition implies necessarily that the monopole and its image are of the same sign. This is simply because the radiated field is unchanged if the actual and virtual sources are permuted, by virtue of symmetry. The interference between the direct and reflected waves is best put in evidence by using the Fourier transform of the pressure, which is thus of the type: e p ( r , r ′; ω ) = iωQ ( ) (ω ) G1 ( r ; ω ) + G2 ( r ′; ω ) [5.211]
(
)
where G1 ( r ; ω ) is the Fourier transform of the Green function of the actual source and G2 ( r ′; ω ) that of its image. With the aid of formula [5.201], where s is replaced
by iω to shift from the Laplace to the Fourier transform, and where t0 = 0 is assumed to simplify notations, the result [5.211] yields: r' − iω ⎛ − iω cr ( e) cf f ρ i ω Q ω ( ) ⎜e e p+ ( r , r ′; ω ) = f ⎜ r + r′ 4π ⎜ ⎝
⎞ (e) ⎟ i ρ f c f kQ (ω ) ⎛ e − ikr e − ikr ' ⎞ + ⎜ ⎟ [5.212] ⎟= r′ ⎠ 4π ⎝ r ⎟ ⎠
where only the outgoing waves are retained. Expression [5.212] can be transformed further by particularizing first the problem to the so-called far field case such that z0 << r , r ′ . Denoting r the length OP and ϕ the angle between OP and P0′P0 , to the first order in the small parameter z0 / r , r and r ′ can be approximated as: ⎛ z ⎞ ⎛ z ⎞ r = r ⎜ 1 − 0 cos ϕ ⎟ ; r ′ = r ⎜ 1 + 0 cos ϕ ⎟ r r ⎝ ⎠ ⎝ ⎠
[5.213]
Furthermore, the difference in the path lengths of the direct and reflected waves is of considerable importance so far as the relative phasing is concerned but its effect on relative magnitude is negligible. Consequently, formula [5.212] can be approximated as: e i ρ f c f kQ ( ) (ω ) e − ikr p+ ( r; ω ) = cos ( kz0 cos ϕ ) [5.214] 2π r or reverting to time domain: p+ ( r; t ) =
e ρ f Q ( ) ( t − r / c f
2π r
) cos
( kz0 cos ϕ )
[5.215]
426
Fluid-structure interaction
The final results [5.214] and [5.215] are remarkable by their relative simplicity. They show that, in the far field approximation, magnitude of the radiated pressure is that which would be radiated in free space by a monopole source twice the strength of the actual one, modulated by a directivity function F (θ ) = cos ( kz0 cos θ ) containing also the factor kz0 which measures the distance of the simple monopoles in terms of the wavelength. In the limiting case z0 → 0 , the result [5.215] becomes “exact” and thus is valid in the near field as well as in the far field domains. Hence, everywhere in the half-space z ≥ 0 , pressure radiated by the monopole is exactly that which would be radiated in free space by a monopole of twice the strength of the actual one, which would located at the same place. This result is often used in practice to enhance the efficiency of volume velocity sources by locating them near a fixed wall or baffle, as further described in subsection 5.3.4.7. The so-called baffled Green function is: ⎧ ρf ⎛ r ⎞ r δ ⎜ t − ⎟ if z ≥ 0 and t ≥ ⎪ ⎧ ρf ⎜ ⎟ c0 if z ≥ 0 ⎪ ⎪ 2π r ⎝ c f ⎠ G+ ( r; ω ) = ⎨ 2π r ⇔ G+ ( r; t ) = ⎨ r ⎪ 0 if z < 0 ⎪ 0 if z < 0 or t < ⎩ ⎪ cf ⎩
[5.216] Of course, such a result can be extended to the case of an impulsion emitted at time t0 and position r0 within the plane z = 0 by using the appropriate time and space shift transformations t → t − t0 and r → r − r0 . 5.3.4.3 3D medium bounded by a pressure nodal plane, dipole sources The results of the last subsection can be transposed to the case of a reflecting plane at which pressure is zero. In practice, this may correspond to a monopole source located near the interface between water and free atmosphere. To fullfil such a boundary condition, it is appropriate that the sign of the image source be opposite to that of the actual source. As could be anticipated, two monopole sources of equal and opposite strength a distance 2 z0 apart from each other form a dipole source, in a similar way as two point forces a distance 2 z0 of equal and opposite strength form a torque. Vanishing of the dipole source when the “lever arm” 2 z0 tends to zero is avoided provided the strength of the monopoles is inversely proportional to 2 z0 instead of being of constant amplitude. Therefore, the monopole strength is written as: a (e) M (e) Q = 2 z0 2 z0
[5.217]
3D Sound waves
427
where a is a length used to scale the strength of the dipole, denoted M ( ) to mark the analogy with a torque, or moment. Therefore with the aid of [5.213], the field [5.212] becomes: e
e i ρ c k M ( ) (ω ) e − ikr p+ ( r , r ′; ω ) = f f 8π rz0
⎛ ⎞ ⎜ e + ikz0 cosϕ ⎟ e − ikz0 cosϕ ⎜ ⎟ − ⎜ ⎛ 1 − z0 cos ϕ ⎞ ⎛ 1 + z0 cos ϕ ⎞ ⎟ ⎟ ⎜ ⎟⎟ ⎜⎜ r r ⎠ ⎝ ⎠⎠ ⎝⎝
[5.218]
which can be written as the rather cumbersome expression: ⎛ z0 cos ϕ ⎞ e cos ( kz0 cos ϕ ) + i sin ( kz0 cos ϕ ) ⎟ i ρ f c f k M ( ) (ω ) e − ikr ⎜⎝ r ⎠ p+ ( r , r ′; ω ) = 2 4π rz0 ⎛ z0 cos ϕ ⎞ 1− ⎜ ⎟ r ⎝ ⎠
[5.219] When z0 tends to zero, the pressure response tends to: e i ρ f c f k M ( ) (ω ) cos ϕ e −ikr p+ ( r, ϕ ; ω ) = p+ ( r , r ′; ω ) = 4π r 2 lim z0 →0
[5.220]
or, reverting to the time domain: p+ ( r,ϕ ; t ) ==
ρ f c f k cos ϕ ∂M ( e ) ( t − r / c0 ) ∂t 4π r 2
[5.221]
It is found convenient to write the forced wave equation corresponding to the field [5.221] by using cylindrical coordinates because of the axial symmetry of the problem which is inherent to a dipole source. e ∂ 2 p 1 ∂p ∂ 2 p 1 ∂ 2 p ∂M ( ) δ ( r ) ∩ δ ′ ( z ) + + − = − ⇔ ρ 0 ∂ r 2 r ∂r ∂z 2 c 2f ∂ t 2 ∂t 2π r
⎧ ∂ 2 p 1 ∂p ∂ 2 p 1 ∂ 2 p + + − =0 ⎪ ∂ r 2 r ∂r ∂z 2 c 2f ∂ t 2 ⎪ ⎨ ρ 0 ∂M ( e ) e ⎪ − = = P( ) (t ) ,0 ; , 0 ; p r t p r t ( ) ( ) + − ⎪ ∂ 2 π r t ⎩
[5.222]
5.3.4.4 Distributed monopole sources and 2D cylindrical waves To derive the Green function in the case of a 2D uniform and isotropic medium, the same method as that used in subsection 5.3.4.1 to obtain the result [5.202] is also
428
Fluid-structure interaction
workable. It consists of writing the forced system [5.197] in terms of cylindrical coordinates, restricted to the case of no axial and circumferential dependency of the solution. Again, to deal with the time dependency of the problem, it is convenient to transform the wave equation by Laplace, or Fourier. Presentation of this method is however postponed to Chapter 7 where it will be worked out in relation to radiative damping problems. Here, we adopt another method which consists of deducing the 2D, and then even the 1D Green functions by integrating the 3D solution [5.202] in the appropriate directions. The presentation given here closely follows that found in [BRU 98] (see also [STA 70]).The 2D case corresponds to a monopole source uniformly distributed along the axis of a cylinder of infinite length. The response to such a line source is obtained by summing the contributions of all the point sources along the line, see Figure 5.27. With this object in mind, the 3D Green function is first reformulated in cylindrical coordinates. The radial distance between the point P of the plane z = 0 and the point source located at the current abscissa z0 is: r = r 2 + z02
[5.223]
The result [5.202] is thus transformed into: G± ( r, z0 ;τ ) =
⎛ r 2 + z02 δ ⎜τ ∓ c0 4π r 2 + z02 ⎜⎝ ρf
⎞ ⎟ where τ = t − t0 ⎟ ⎠
[5.224]
Figure 5.27. Integration of the line source contribution to the pressure in the plane z = 0
Therefore the 2D Green function is given by the integral along the source line:
3D Sound waves
429
+∞
G± ( r;τ ) =
⌠ ⎮ ⎮ ⎮ ⎮ ⎮ ⌡−∞
⎛ r 2 + z02 δ ⎜τ ∓ cf 4π r 2 + z02 ⎜⎝ ρf
⎞ ⎟ dz0 ⎟ ⎠
[5.225]
To carry out the integration, it is however more convenient to use the variable r than z0 . Using formula [5.223], we notice that: dz0 dr = r z0
[5.226]
Thus the integral [5.225] is rewritten as:
G± ( r;τ ) =
⎛ r ⎞ +∞ ⌠ ⎛ ⎛ r ⎞ r ⎞ ⎜ ⌠⎮ ⎟ ⎮ ∓ ∓ δ τ δ τ ⎜⎜ ⎟⎟ ⎜⎜ ⎟⎟ ⎜⎮ ⎟ ⎮ c c f ⎠ f ⎠ ⎝ ⎝ ⎜ ⎮⎮ dr + ⎮⎮ dr ⎟ ⎜ ⎮ − r 2 − r2 ⎟ ⎮ r 2 − r2 ⎮ ⎜ ⎮⎮ ⎟ ⎮ ⌡r ⎜ ⌡−∞ ⎟ ⎝ ⎠
+∞
⎛ r ⎞ δ ⎜τ ∓ ⎟ ⎜ c f ⎟⎠ ρ ⎝ dr = f z0 4π
+∞
⎛ ⎛ r ⎞ r ⎞ δ ⎜τ ∓ ⎟ U ⎜τ ∓ ⎟ ⎜ ⎟ ⎜ cf ⎠ c f ⎟⎠ ρ c ⎝ ⎝ dr = f f 2 2π r 2 − r2 (τ c f ) − r 2
⌠ ⎮ ρ f ⎮⎮ ⎮ 4π ⎮ ⎮ ⎮ ⌡−∞
and finally as:
G± ( r;τ ) =
⌠ ⎮ ρ f ⎮⎮ ⎮ 2π ⎮ ⎮ ⎮ ⌡r
[5.227]
Where once more, the Heaviside step function marks fulfilment of the principle of causality. An interesting feature of the Green function [5.227] is that, in contrast with the 3D case, though the source varies in time as a Dirac pulse, the induced pressure field is not a travelling pressure pulse but a wave, which starts at t = t0 + r / c f and then tends to zero as 1/t. The method adopted here to deduce the Green function [5.227] is well suited to understand such a behaviour, as it points out that all the source monopoles are contributing to the pressure wave observed at the distance r from the origin of the z = 0 plane. The arrival time of the waves emitted by the monopoles located at ± z0 is t = t0 + r 2 + z02 / c f where z0 extends from 0 to ∞ . 5.3.4.5 Distributed monopole sources and plane waves The method described in the last subsection can be further extended to deduce the 1D Green function in a compact form. To the point source in the 1D space corresponds a plane source in the 3D space, normal to the axis selected as the 1D
430
Fluid-structure interaction
space, for instance Ox. The 2D Green function [5.227] is first expressed in Cartesian coordinates as: G± ( x, y;τ ) =
(
2 2 ρ f cf U τcf ∓ x + y
2π
(τ c ) − ( x 2
f
2
)
[5.228]
+ y2 )
As indicated in Figure 5.28, the 1D Green function can be obtained by letting the the line source Oz sweep the whole plane Oyz. In other terms, it can be calculated by integrating the 2D Green function in the Oy direction. The result is: +y
G ( x ;τ ) =
ρ f cf
where y1 =
2π
⌠ 1 ⎮ ⎮ ⎮ ⎮ ⌡− y1
(τ c ) f
2
+ y1
ρ c ⎡ ⎛ y ⎞⎤ = f f ⎢sin −1 ⎜ ⎟ ⎥ 2 2 2π ⎣ y1 − y ⎝ y1 ⎠ ⎦ − y1 dy
[5.229]
− x 2 . Hence the final result where the incoming and outgoing
waves are distinguished: G± ( x;τ ) =
ρ f cf 2
⎛ x⎞ U ⎜τ ∓ ⎟ c0 ⎠ ⎝
[5.230]
Figure 5.28. 3D Plane source resulting in a 1D point source
Thus it is found that the Dirac pulse triggers a travelling pressure wave which starts at t = t0 + x / c f and which remains constant afterwards, as could be expected for a plane wave and best demonstrated by performing the integration over the plane
3D Sound waves
431
source in a little more subtle way than just above. Refering to Figure 5.27, it is noticed that, as a consequence of the reciprocity theorem, the 1D Green function can be obtained as the axial pressure at abscissa z0 resulting from the integration of the 3D Green function [5.224] over the plane z = 0. The integral being carried out in polar coordinates, it becomes obvious that the 1/r decrease of the spherical pressure wave is precisely compensated for by the increase of the elementary area of integration dS = 2π rdr , so the resulting pressure is a Heavyside step function. Retaining the outgoing wave, the integration is conducted as follows: +∞
G± ( z0 ;τ ) =
⌠ ⎮ ⎮ ρf ⎮ ⎮ 4π ⎮⎮ ⎮ ⎮ ⌡0
⎛ r 2 + z02 δ ⎜τ − ⎜ cf ⎝ r 2 + z02
⎞ ⎟ ⎟ ⎠ 2π rdr = ρ f 2
+∞
⌠ ⎮ ⎮ ⎮ ⎮ ⎮ ⌡z0
⎛ r ⎞ δ ⎜ τ − ⎟ dr ⎜ c f ⎟⎠ ⎝
[5.231]
where r = r 2 + z02 ⇒ rdr = rdr . The incoming wave would be accounted for simply by changing the sign of z0 . Hence the 1D Green function along the Oz axis: G± ( z0 ;τ ) =
ρ f cf 2
⎛ z ⎞ U ⎜τ ∓ 0 ⎟ ⎜ c f ⎟⎠ ⎝
[5.232]
5.3.4.6 Distributed monopole sources and first Rayleigh integral
Figure 5.29. Sound emission from monopole sources distributed within a finite volume
Considering now a velocity source distributed within a finite volume (V ) e according to the density law Q ( ) ( r0 ; t ) , see Figure 5.29. The response at a point r
432
Fluid-structure interaction
outside (V ) is obtained by using the superposition principle as the following integral: ⌠
⎮ p( r ; t ) = − ρ f ⎮⎮
Q ( e ) ( r0 ; t − r / c f ) U ( t − r / c f
⎮ ⌡(V )
4π r
)dr
0
[5.233]
Particularisation of the integral [5.233] to the case of a source distributed over a surface (S ) instead of a volume is immediate, as it suffices to restrict the integration domain on (S ) . The result is broadly known as the Rayleigh integral (see for instance [PIE 91], [BLA 00]): ⌠ ( e ) ( r ; t − r / c ) U ( t − r / c ) ⎮ Q f f 0 p( r ; t ) = − ρ f ⎮⎮ dr0 4 π r ⎮
[5.234]
⌡(S )
Unfortunately, calculation of the integral [5.234] is generally difficult to carry out analytically, except for a few problems. Amongst them, the case of the acoustic radiation from a circular piston is a particularly entlightening and well documented example, which is the subject of an introductory presentation in the next subsections. More detailed information on the subject is available in the published literature which is particularily abundant on the subject. The interested reader can be refered in particular to [BLA 00]). 5.3.4.7 Pressure field in the axial direction by a baffled circular piston As shown in Figure 5.30, where Oxyz stands for a Cartesian frame, the piston of centre O and radius R0 is assumed to lie in the Oxy plane and vibrate in the transverse direction Oz according to the uniform displacement law Z = Z 0 eiω t . Furthermore, the piston is assumed to be embedded in a coplanar and fixed baffle as also shown in Figure 5.30. The purpose of the baffle is to increase the source efficiency by restricting the sound field to the postitive domain z ≥ 0 . One is interested in calculating the pressure field induced at the field position P by the piston motion. As the source is baffled, the Rayleigh integral [5.234] must be multiplied by two, in agreement with the results of subsection 5.3.4.2. On the other hand, by virtue of the symmetry of the problem about the Oz axis, we can assume that P lies in the Oxz plane. Thus, in the present case, the Rayleigh integral takes the form:
3D Sound waves
2
−ω ρ f Z 0 e p( r ; t ) = 2π
⌠
iω t ⎮
−
⎮ ⎮ ⎮ ⎮ ⌡(S )
e
iω r cf
U (t − r / c f r
)dr
0
433
[5.235]
Figure 5.30. Circular baffled piston
For the sake of simplicity, it is found apposite to start by working out the integral [5.235] in the particular case of a point receptor located on the Oz axis. Calculation can proceed in line with that already performed in subsection 5.3.4.4. Actually, it suffices to adapt the boundaries of the integral [5.231] to the present geometry: ⌠
p( z; t ) = −ω 2 ρ f Z 0 eiω t ⎮⎮
R02 + z 2
⌡z
e-ikr dr
[5.236]
which is readily integrated to yield the axial pressure field, expressed as: ⎛ i ω t − kz ) i (ω t − k p( z; t ) = P0 ⎜ e ( −e ⎝
R02 + z 2
)⎞ ⎟ ⎠
[5.237]
where the pressure P0 used as a convenient scaling factor is defined as: P0 = iωρ f c f Z 0
[5.238]
So defined P0 is an imaginary quantity, which means that it is in quadrature of phase with the piston displacement. On the other hand, the pressure field [5.237] is the sum of two plane waves of the same magnitude. The first one is termed the direct wave as it seems to be emitted directly at the centre of the piston and the
434
Fluid-structure interaction
second is termed the edge wave as it seems to be emitted at the edge of the disk. These two components are out-of-phase with each other. As an asymptotic result, if the radius of the disk tends to infinity, the edge wave is delayed indefinitely and only the direct wave remains, as should be. Combination of the two plane waves results in a spherical wave in the far field, due to the phase difference between them. If z/ R0 is sufficiently large, the wave [5.237] can be approximated as: p( z; t ) = P0 e (
i ω t − kz )
− ikz ( ⎛ ⎜1 − e ⎝
) ⎞ P ei (ω t −kz )
R02 / z 2 +1 −1
⎟ ⎠
0
(1 − e
− ikR02 / 2 z
)
[5.239]
The exponential term let appear as scaling length, the so-called Rayleigh’s distance, denoted ΛR , which is appropriate to mark, as an order of magnitude at least, the transition from the near to the far field component in the pressure signal [5.237]: ΛR =
kR02 ω R02 = 2 2c f
[5.240]
Hence, if z is much larger than the Rayleigh distance, expression [5.239] can be safely approximated by expanding the exponential up to the first order, which results in the spherical wave: p ( z; t )
2 iP0 ΛR i (ω t − kz ) −ω ρ f S0 Z 0 i (ω t − kz ) e = e z 2π z
[5.241]
where S0 = π R02 is surface area of the disk. It is also of interest to describe the major features of the pressure field including the nearfield range. Actually, in the range 0 ≤ z ≤ ΛR , the interfering waves result in a series of peaks and nulls, the number of which depends on the ratio of the disk radius to the sound wavelength, as brought in evidence by rewriting formula [5.237] as follows: p( z; t ) = 2iP0 sin ( k Δ z ) e (
i ω t −k ( z + Δ z ))
[5.242]
where Δ z is half the difference in the sound paths of the two waves, defined as: 2Δ z = R02 + z 2 − z
[5.243]
The interference pattern characterized by the function sin ( k Δ z ) is illustrated in Figure 5.31, in which the modulus of the pressure divided by the scale factor P0 is plotted versus the axial distance from the disk, divided by the Rayleigh distance. The far field approximation is plotted in dashed line.
3D Sound waves
435
(a)
(b) Figure 5.31. Axial pressure field in reduced coordinates (a): large disk, (b) small disk
The plots illustrate two distinct cases. In (a), the radius of the disk is larger than the sound wavelength. As a consequence, the nearfield ( z < ΛR ) is marked by a series of peaks and nulls, the number of which depends on the ratio R0 / Λ . In (b), the radius of the disk is smaller than the sound wavelength and the nearfield tends asymptotically to the plateau value without any interference oscillations, as z tends to zero. In both cases, the far field asymptote is practically reached for z larger than one up to a few ΛR .
436
Fluid-structure interaction
5.3.4.8 Directivity of sound radiated by a baffled circular piston If the receptor point is offset from the Oz axis, the calculation can still be carried out analytically, in the far field approximation at least.
Figure 5.32. Receptor point offset from the Oz axis and source point in the Oxy plane
Denoting ϕ the angle between OP and Oz (see Figure 5.32), the distance between the receptor point P, assumed to lay in the Oxz plane and a current source point P0 in the plane Oxy is obtained as: r − r0
2
= ( r sin ϕ − r0 cos θ ) + r 2 cos2 ϕ + r02 sin 2 θ 2
[5.244]
Hence: r = r − r0 = r 2 + r02 − 2rr0 sin ϕ cos θ
[5.245]
The far field approximation of [5.245] is written as: r r − r0 sin ϕ cosθ
[5.246]
As in the case of the pair of monopoles of the same sign (cf. subsection 5.3.4.2), and for the same reason, the difference in the path lengths between the contribution of two point sources of the disk surface is of considerable importance as far as the relative phasing is concerned but its effect on the relative magnitude is negligible. So the far field approximation of the Rayleigh integral [5.235] is written as: p( r, ϕ ; t ) =
−ω 2 ρ0 Z 0 e ( 2π r
i ω t − kr )
R
2π
⌠ 0 ⌠ ⎮ r0 dr0 ⎮ ⎮ ⎮ ⌡0 ⌡0
e(
i - kr0 sin ϕ cosθ )
dθ
[5.247]
3D Sound waves
437
The integral in θ is known as: 2π
⌠ ⎮ ⎮ ⌡0
e(
i - kr0 sin ϕ cosθ )
dθ = 2π J 0 ( kr0 sin ϕ )
[5.248]
Then, integration in r0 proceeds as follows: p( r, ϕ ; t ) =
−ω 2 ρ f Z 0 ei ( ωt − kr ) r
R
⌠ 0 ⎮ r0 J 0 ⎮ ⌡0
( kr0 sin ϕ ) dr0
[5.249] i ω t − kr )
Letting u = kr0 sin ϕ we obtain p( r, ϕ ; t ) =
−ω 2 ρ f Z 0 e (
r ( k sin ϕ )
2
kR0 sin ϕ
⌠ ⎮ ⌡0
uJ 0 ( u ) du and the
far field pressure field is finally expressed as: ⎛ −ω 2 ρ f R0 Z 0 ei ( ωt -kr ) ⎞ ⎛ J 1 ( kR0 sin ϕ ) ⎞ p( r, ϕ ; t ) = ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ sin ϕ r ⎠ ⎝ ⎠
[5.250]
The term within the first brackets stands for a spherical wave radiated by a monopole source concentrated at O, and the term within the second bracket is an angular weighting factor which can be used to define an amplitude directivity factor: D (ϕ ) =
p( r,ϕ ; t ) 2 J 1 ( kR0 sin ϕ ) = p( r,0; t ) kR0 sin ϕ
−
π π ≤ϕ ≤ + 2 2
[5.251]
As illustrated in Figure 5.33, the radiation pattern of the baffled circular piston becomes highly directive as soon as the radius of the disk becomes larger than the sound wavelength. Maximum pressure is in the normal direction, as could be expected. Incidentally, as barely visible in the bottom polar plot of Figure 5.33, if R0 / Λ is larger than unit, the main lobe is flanked by side lobes of much smaller amplitude. 5.3.4.9 Dipole radiation by the unbaffled circular piston integral equation (KH) The pressure field excited by an unbaffled disk vibrating in the normal direction of its own plane (assumed to be Oz direction as in the last subsection), is dipolar in nature since at the same time the fluid is pushed by a face of the disk, it is pulled by the other. As a consequence, a gradient of pressure develops in the Oz direction and in the absence of baffle the fluid undergoes a reciprocating motion from one face of the disk to the other, near the edge. In the low frequency range, the oscillating flow is almost incompressible, hence practically no sound is radiated. However, as frequency is enhanced, the importance of fluid compressibility also increases and efficiency of sound radiation is improved. Such a qualitative scenario was proposed
438
Fluid-structure interaction
Figure 5.33. Directivity patterns of sound radiation for the circular baffled piston
first by Stokes [STO 68]. More generally, a similar qualitative behaviour holds for any vibrating rigid bodies immersed in an infinite extent of fluid. Quantitative analysis for the disk is postponed to the next subsection as an application of the Kirchhoff-Helmholtz integrals introduced just below.
3D Sound waves
5.3.5
439
Weighted integral formulations
As in structural mechanics, in acoustics also it can be very fruitful to formulate the problem by using a weighted integral instead of a differential equation. This is made possible by performing the functional scalar product of the local equation by a weighting functional vector W (r ; t ) judiciously selected. 5.3.5.1 The Kirchhoff-Helmholtz integral theorem Considering a uniform fluid initially at rest, the local wave equation forced by an external acoustic source is written in terms of pressure as: 1 e Δp( r ; t ) − 2 p( r ; t ) = S ( ) ( r ; t ) cf p( r ;0) ≡ 0 ; grad p( r ;0) ≡ 0
where S (
e)
(r;t )
r ∈ (V )
[5.252]
stands for the volume density of the source, whatever its nature
may be. Furthermore, equation [5.252] is supposed to hold within a finite volume (V ) , bounded by a closed surface (S ) .To get rid of the local character of such an equation, it is suitable to project it onto a functional vector W ( r ; t ) . Going a step further, projection can also be extended to the time axis, which yields the following integral equation: T
⌠ ⎮ ⎮ ⎮ ⎮ ⌡0
⎧⌠ 1 ⎛ ⎞ ⎪⎮ p ( r ;τ ) ⎟ dV ⎨ ⎮ W ( r ;τ ) ⎜⎜ Δp( r ;τ ) − 2 ⎟ cf ⎪⎩ ⎮⌡(V ) ⎝ ⎠
T ⎫ ⌠ ⎧⌠ ⎫⎪ ⎪ ⎮ ⎪⎮ (e) ⎬ dτ = ⎮ ⎨ ⎮ W ( r ;τ ) S ( r ;τ ) ⎬ dτ ⎮ ⎪ ⌡(V ) ⎪⎭ ⎭⎪ ⌡0 ⎩
[5.253] where T = t + ε is a time larger than t which will disappear in the final result. Let us assume that the volume source density S ( ) is nul, at a first step at least. It is not a difficult task to show that the adjoint form of [5.253] is: e
T
⌠ ⎮ ⎮ ⎮ ⎮ ⎮ ⎮ ⌡0
⌠ ⎮ ⎮ ⎮ ⎮ ⎮ ⌡(V )
⎛ 1 ⎞ 1 p ⎜ ΔW − 2 W ⎟⎟ dV dτ − 2 ⎜ c c f f ⎝ ⎠
⌠ ⎮ ⎮ ⎮ ⌡(S )
τ =0 (W grad p − pgradW ) . ndSd
⌠ ⎮ ⎡⎣ pW ⎮ ⎮ ⌡(V )
0
[5.254]
T
⌠ ⎮ ⎮ ⎮ ⎮ ⎮ ⌡0
T
⎤⎦ dV + − pW
440
Fluid-structure interaction
To establish [5.254], the compressibility term is integrated by parts twice and the inertial term is written in Cartesian coordinates using indicial notations (cf. [AXI 05], Chapter 3):
X WΔp dV = XY W ∂ FG ∂ p IJ dV YZa f YY ∂ x H ∂ x K Za f V
V
i
[5.255]
i
Then, form [5.255] is integrated by parts twice to yield the adjoint form:
X WΔp dV = XY FG W ∂ p − p ∂ W IJ dS + XY p ∂ FG ∂ W IJ dV YZa f YY H ∂ x ∂ x K YY ∂ x H ∂ x K Za f Z V
i
a Sf
i
i
V
[5.256]
i
Reverting to the symbolic tensor notation, it is finally written as: ⌠ ⎮ W Δp ⎮ ⌡(V )
⌠
dV = ⎮⎮
⎮ ⌡(S )
+ (W grad p − pgradW ) .ndS
⌠ ⎮ ⎮ ⌡(V )
pΔW dV
[5.257]
As expected, owing to its conservative nature, the Laplacian is found to be formally self-adjoint. The point of major interest of the adjoint form [5.254] is to let the time and space boundary terms of the problem appear explicitly. Furthermore, if the Green function in uniform and open space [5.202] is selected as the weighting function, volume and time integrations can be carried out immediately, due to the concentrated nature of the source. Actually, it is prefered to devide the Green function [5.202] by ρ f to obtain directly an appropriate physical dimensioning of the quantities of interest. Using equation [5.197], the first integral in [5.254] reads as: T
I1 =
⌠ ⎮ ⎮ ⎮ ⎮ ⎮ ⎮ ⎮ ⌡0
⌠ ⎮ ⎮ ⎮ ⎮ ⎮ ⌡(V )
⌠
⎮ 1 ⎛ 1 ⎞ p ( r0 ; t0 ) ⎜⎜ ΔG − 2 G ⎟⎟ dV dt0 = − ⎮⎮ ρf ⎝ cf ⎠ ⎮ ⎮ ⌡
T
0
⌠ ⎮ ⎮ ⌡(V )
p ( r0 ; t0 ) δ ( r − r0 )δ (t − t0 )dr0 dt0
[5.258] Since r0 lies within (V ) and T is supposed greater than t, the right-hand side of [5.258] reads as: I1 = − p ( r ; t ) [5.259]
3D Sound waves
441
Both the initial and final terms included in the second volume integral of [5.254] vanish. The initial terms vanish because the medium is initially at rest, so p( r ;0) and p (r ;0) are zero. The final terms vanish because T is supposed larger than t: G ( r , r0 ; t − T ) ≡ 0 ; G ( r , r0 ; t − T ) ≡ 0 ∀ t < T
[5.260]
With the aid of [5.258] to [5.260] the adjoint form [5.254] takes on the remarkable form broadly known as the Kirchhoff–Helmholtz integral equation (in short notation K.H. integral): t
p( r ; t ) =
⌠ ⎮ ⌠ 1 ⎮ ⎮ ⎮ dt0 ⎮ ⎮ ρf ⎮ ⌡(S ) ⎮ ⌡0
(Ggrad p − pgradG ).ndS
[5.261]
where in the upper boundary of the time integral ε > 0 is let vanish. According to the K.H. integral, the pressure at any receptor point within the fluid can be determined once the pressure and the normal component of the pressure gradient (hence the normal velocity of the fluid particles) is known at each point of the boundary. In other and maybe less abstract terms, to prescribe the pressure over a part (S1 ) of the fluid boundary (S ) is equivalent to excite the medium by a pressure source distributed over
(S1 ) ,
while prescribing the normal gradient of
pressure is equivalent to excite the medium by a volume velocity source. Generally, the problem to be solved mixes both kinds of sources. However, as clarified later, pressure and normal gradient of pressure on (S ) cannot be prescribed as independent source fields. In particular, pressure has the so-called duplication property of appearing both in the right-hand and the left-hand side of equation [5.261]. 5.3.5.2 Particularization of K.H. integral to plane waves As a first exercise to solidify the abstract formalism presented just above, we particularize the Kirchhoff-Helmholtz integral [5.261] to the case of plane waves in a uniform tube of cross-sectional area S f and length L, which are excited by transient sources concentrated at one end of the tube. As the present problem is onedimensional, the weighted integral [5.254] can be written as:
442 T
⌠ ⎮ ⎮ ⌡0
Fluid-structure interaction L
⌠ ⎮ ⎮ ⎮ ⌡0
⎛ ∂ 2W 1 ∂ 2W ⎞ − p( x0 ; t0 ) ⎜ dt dx = ⎜ ∂ x 2 c 2 ∂ t 2 ⎟⎟ 0 0 f 0 0 ⎠ ⎝
T
⌠ ⎮ ⎮ ⎮ ⎮ ⎮ ⌡0
L
⎡ ∂W ∂ p⎤ −W ⎢p ⎥ dt0 − x x0 ⎦ 0 ∂ ∂ 0 ⎣
L
⌠ 1 ⎮⎮ ⎮ c 2f ⎮ ⎮ ⌡0
T
[5.262]
⎡ ∂W ∂ p⎤ −W ⎢p ⎥ dx0 t t0 ⎦ 0 ∂ ∂ 0 ⎣
where we interchange the source and observation points, which leaves the Green function unchanged. As expected, the right-hand side of equation [5.262] comprises two distinct terms, namely the time integral involving the conditions which hold at the space boundaries and the space integral involving the conditions which hold at the time boundaries. It is emphasized that contribution of the boundary conditions would vanish if the weighting function would be selected so as to comply with the boundary conditions of the problem. Actually, if it were the case, the problem would be formulated as a self-adjoint system. Nevertheless, the object of the KirchhoffHelmoltz integral is precisely the reverse, to solve the problem in terms of conditions at the boundaries which act as external sources, solely. With this object in mind, it is appropriate to select as the weighting function the one-dimensional Green function [5.230] devided by the fluid density. Equation [5.262] is thus transformed into: T
⌠ ⎮ ⎮ ⎮ ⎮ ⎮ ⌡0
L
⎡ ∂p 1 ∂G⎤ −p ⎢G ⎥ dt0 − 2 ∂ x0 ⎦ 0 cf ⎣ ∂ x0
L
⌠ ⎮ ⎮ ⎮ ⎮ ⎮ ⌡0
T ⎧ p ( r ; t ) if T > t ⎡ ∂ p ∂ G⎤ − = G dx ⎨ 0 ⎢ ⎥ if T > t ⎩0 ⎣ ∂ t0 ∂ t0 ⎦ 0
[5.263]
Once more, as it refers to the response of a medium initially at rest, the contribution of the initial conditions vanishes automatically. Contribution of the final conditions are shown to depend on the time interval selected for integration. It is found expedient to suppose that T is larger than t to let vanish the final conditions. On the other hand, it is recalled that, in terms of distributions, the derivative of the Heaviside step function is the Dirac impulsion (cf. [AXI 04]): dU = δ (ϑ ) dϑ
[5.264]
Therefore G and its derivatives are nil at t = 0 and t0 = t + ε where ε is assumed to be positive. So, the contribution of the final conditions to the solution is also zero. The first term in the left-hand side reduces to the pressure field p ( x; t ) to be determined in terms of the surface sources concentrated at the boundaries of the medium. As a preliminary, the Green functions and derivatives are expressed as:
3D Sound waves
G+ ( x − x0 ; t − t0 ) = G− ( x − x0 ; t − t0 ) =
ρ f cf 2 ρ f cf 2
443
⎛ x − x0 ⎞ U ⎜ t − t0 − ⎟ if x > x0 ⎜ c f ⎟⎠ ⎝ ⎛ x −x⎞ U ⎜ t − t0 − 0 ⎟ if x < x0 ⎜ c f ⎟⎠ ⎝
[5.265]
x − x0 ⎞ ∂ G+ ρ f ⎛ = δ ⎜ t − t0 − ⎟ if x > x0 2 ⎜⎝ c f ⎟⎠ ∂ x0 ρ ⎛ x −x⎞ ∂ G− = − f δ ⎜ t − t0 − 0 ⎟ if x < x0 2 ⎜⎝ c f ⎟⎠ ∂ x0
Let us consider first the waves triggered by prescribing the pressure at the tube ∂p ends. This case correspond to the conditions p 0, L ≠ 0 and ≡ 0 . The ∂ x0 0, L contribution of the boundaries to the solution reads as: T
⌠ ⎮ 1 ⎮⎮ ρ f ⎮⎮ ⎮ ⌡0
⎛ ∂ G+ ⎞ ⎛ ∂ G− ⎞ −⎜ p dt0 = ⎜p ⎟ ⎟ ⎝ ∂ x0 ⎠ x0 =0 ⎝ ∂ x0 ⎠ x = L 0
T
1 2 1 2
⌠ ⎮ ⎮ ⎮ ⎮ ⎮ ⌡0
⎧⎪ ⎛ x ⎨ p(0; t0 )δ ⎜⎜ t − t0 − cf ⎪⎩ ⎝
⎛ x p ⎜ 0; t − ⎜ c f ⎝
⎞ 1 ⎟⎟ + ⎠ 2
⌠
⎞ ⎫⎪ 1 ⎮⎮ ⎟⎟ ⎬ dt0 + ⎮ 2⎮ ⎠ ⎪⎭ ⎮
T
⌡0
⎧⎪ ⎛ L − x ⎞ ⎫⎪ ⎟ ⎬dt0 = ⎨ p(0; t0 )δ ⎜⎜ t − t0 − c f ⎟⎠ ⎪⎭ ⎪⎩ ⎝
[5.266]
⎛ L−x⎞ p ⎜ L; t − ⎟ ⎜ c f ⎟⎠ ⎝
Considering then the waves triggered by prescribing the particle velocity at the ∂p tube ends, we adopt the conditions p 0, L ≡ 0 and ≠ 0 , we find: ∂ x0 0, L T
⌠ ⎮ 1 ⎮ ⎮ ρf ⎮ ⎮ ⌡0
⎛ ⎛ ∂p ⎞ ∂p ⎞ − ⎜ G+ dt0 = ⎜ G− ⎟ ⎟ ⎝ ∂ x0 ⎠ x0 = L ⎝ ∂ x0 ⎠ x0 =0
[5.267]
T
⌠ c f ⎮⎮ ⎮ 2 ⎮ ⎮ ⌡0
⎪⎧ ∂ p ⎨ ∂x ⎩⎪ 0
⎛ L− x⎞ ∂ p ⎜⎜ U (t − t0 − ⎟− c f ⎟⎠ ∂ x0 x0 = L ⎝
⎛ x ⎜⎜ U (t − t0 − cf x0 = 0 ⎝
⎞ ⎫⎪ ⎟⎟ ⎬ dt0 ⎠ ⎭⎪
444
Fluid-structure interaction
Whence the wave: T−
⌠ c f ⎮⎮ 2 ⎮⎮ ⎮ ⌡0
L− x cf
T−
∂p ∂ x0
dt0 − x0 = L
x
⌠ cf c f ⎮⎮ 2 ⎮⎮ ⎮ ⌡0
∂p ∂ x0
[5.268]
dt0 x0 = 0
The pressure field resulting from a compound boundary source system is obtained by applying the superposition principle. Thus it is of the general form: ⎛ ⌠ t − Lc− x ⎜⎮ f ∂ p ⎛ ⎛ L− x⎞ x ⎞ + − + p t c 2 p ( x; t ) = p ⎜ L; t − 0; ⎟⎟ ⎜⎜ ⎟⎟ f ⎜ ⎮⎮ ⎜ c c ∂ x0 ⎜ ⎮⎮ f f ⎠ ⎝ ⎠ ⎝ ⎜ ⌡0 ⎝
t−
x
⌠ cf ⎮ dt0 − ⎮⎮ ⎮ x0 = L ⎮ ⌡0
⎞ ⎟ ∂p dt0 ⎟ ∂ x0 x = 0 ⎟ 0 ⎟ ⎠
[5.269] Furthermore, it can be shown that the remaining integrals can also be solved. The key point is to use the duplication property of pressure mentioned at the end of the last subsection. Thus the form [5.269] is applied at the boundaries and at successive time intervals which fit to the back and forth travel time of the waves. Substituting x = L and then x = 0 into [5.269], we obtain in the interval 0 ≤ t ≤ L / cf : t
p ( L; t ) =
⌠ ⎮ ∂p c f ⎮⎮ ∂ x0 x = L ⎮ ⎮ 0 ⌡0
t
;
dt0
p ( 0; t )
⌠ ⎮ ∂p = c f ⎮⎮ dt0 ∂ x0 x = 0 ⎮ ⎮ 0 ⌡0
[5.270]
Then, it is realized that the field p(x;t) can be expressed according to a series of retarded waves, which is what we must expect due to the reflective properties of the tube ends chosen to illustrate the method. At this step, it is also found worthwhile to address the few following points, as additive notes to the presentation of the Kirchhoff-Helmholtz integral. NOTE 1
– Solution of the K.H. integral and time interval of integration
It is of interest to show that the same solution as [5.269] is obtained if the weighted form is integrated in a slightly restricted time interval [0,t-ε]. The time derivative of the Green function [5.265] is: x − x0 ∂ G cf ⎛ = δ ⎜ t − t0 − 2 ⎜ cf ∂ t0 ⎝
⎞ ⎟ ⎟ ⎠
Hence the term of the final conditions of motion is written as:
3D Sound waves
445
L
⌠ 1 ⎮⎮ ⎮ 2c f ⎮ ⎮ ⌡0
⎧⎪ ⎨U ⎩⎪
⎛ x − x0 ⎜⎜ ε − cf ⎝
⎞∂ p ⎛ x − x0 + p(t − ε ; x0 )δ ⎜ ε − ⎟⎟ ⎜ cf ⎠ ∂ t0 ⎝
⎞ ⎫⎪ ⎟⎟ ⎬ dx ⎠ ⎭⎪
0
The contribution of the step function is found to be: x
L
⌠ ⎮ 1 ⎮ ⎮ 2c f ⎮ ⎮ ⎮ ⌡0
⎧⎪ ⎨U ⎩⎪
⎛ x − x0 ⎞ ∂ p ⎫⎪ 1 ⎜ε − ⎟ dx0 + ⎬ ⎜ c f ⎟ ∂ t0 ⎪ 2c f ⎝ ⎠ ⎭
⌠ ⎮ ⎮ ⎮ ⎮ ⎮ ⎮ ⌡x
⎧⎪ ⎨U ⎩⎪
⎛ x − x0 ⎜ε + ⎜ cf ⎝
⎞ ∂ p ⎫⎪ ⎟ dx ⎟ ∂ t0 ⎬⎪ 0 ⎠ ⎭
That is: ⌠
x
1 ⎮ 2c f
⎮ ⎮ ⎮ ⌡x −ε c0
∂p ∂ t0
⌠
dx0 +
x +ε c0
1 ⎮ 2c f
⎮ ⎮ ⎮ ⌡x
∂p ∂ t0
⌠
dx0
=
x +ε c0
1 ⎮ 2c f
⎮ ⎮ ⎮ ⌡x −ε c0
∂p ∂ t0
dx0
Obviously it vanishes if ε tends to zero. The contribution of the Dirac pulse is written as: x
⌠ ⎮ 1 ⎮ ⎮ 2c f ⎮ ⎮ ⌡0
L
⎛ x − x0 ⎞ 1 δ ⎜ε − ⎟ p(t − ε ; x0 ) dx0 + ⎜ ⎟ c 2 cf f ⎝ ⎠
⌠ ⎮ ⎮ ⎮ ⎮ ⎮ ⌡x
⎛
δ ⎜ε
⎜ ⎝
−
x − x0 ⎞ ⎟ p(t − ε ; x0 ) dx0 c f ⎟⎠
Recalling that: b
⌠ ⎮ δ (α x )dx ⎮ ⌡a
⎧1 ⎪ if 0 ∈ [ a , b] = ⎨α ⎪⎩ 0 otherwise
we arrive at: x
⌠ ⎮ 1 ⎮ ⎮ 2c f ⎮ ⎮ ⌡0
⎛ x − x0 ⎞ 1 δ ⎜ε − ⎟⎟ p(t − ε ; x0 ) dx0 = p(t − ε ; x − ε c0 ) ⎜ c 2 f ⎝ ⎠
L
⌠ ⎮ 1 ⎮ ⎮ 2c f ⎮ ⎮ ⌡x
⎛ x − x0 ⎞ 1 δ ⎜ε − ⎟⎟ p(t − ε ; x0 ) dx0 = p(t − ε ; x + ε c0 ) ⎜ cf ⎠ 2 ⎝
If ε tends to zero, the above expression tends to p(x;t), as should be. NOTE 2
– Sources distributed within the fluid column
446
Fluid-structure interaction
In presence of acoustic sources distributed within the whole volume of the fluid column, it is appropriate to add to the preceding terms the volume contribution: p ( x; t ) =
1 ρf
t
⌠ ⎮ ⎮ ⌡0
L
⌠ ⎮ ⎮ ⎮ ⌡0
S ( ) ( x0 ; t0 )G ( x − x0 ; t − t0 )dx0 dt0 e
[5.271]
Formula [5.271] reads as a convolution product carried out in the time-space domain, as already explained in Chapter 3 subsection 3.2.4.1. Going a step further, if a Green function satisfying the homogeneous boundary conditions of the problem is selected instead of the 1D open-space Green function [5.265], the acoustic response is entirely described by the volume contribution [5.271], as should be. NOTE 3
– Response in the spectral domain
In many problems of practical interest, it is found more appropriate to work in the spectral than in the time domain. Of course, to shift from time to frequency domain the Fourier transform is used: ⌠ p ( x; ω ) = ⎮⎮
+∞
⌡−∞
p ( x; t ) e − iωt dt
The Fourier transform of the Green function [5.265] is written as: x − x0 x − x0 ρ f c f − iω c f ∂ G ρ f −iω c f ; G ( x − x0 ; ω ) = − e ( x − x0 ; ω ) = ± e 2iω 2 ∂ x0
[5.272]
Whence the pressure field in absence of volume source terms: ( L− x) x c ∂ p c ∂ p ⎞ − iω ⎞ − iω 1⎛ 1⎛ p ( x; ω ) = ⎜ p ( 0; ω ) − f ( 0; ω ) ⎟ e c f + ⎜ + p ( L; ω ) + f ( L; ω ) ⎟ e c f 2⎝ 2⎝ iω ∂ x0 iω ∂ x0 ⎠ ⎠
[5.273] 5.3.5.3 Application to plane acoustic waves triggered by a transient The problem is depicted in Figure 5.34. The piping system consists of a straight tube of length L and constant cross-sectional area S f , which is connected at one end to a large water tank, and provided with a sluice gate at the other end. The gate is assumed to be open initially. As a consequence, there is a steady outflow of water at constant volume velocity, denoted Q0 . At time t = 0, the gate is actioned to close the tube. Some time is necessary to stop completely the flow. To simplify the analytical calculations, without changing the physics of the problem, the volume velocity during the closing time is described as the idealized transient:
3D Sound waves
⎛ t ⎞ Q ( t ) = Q0 ⎜ 1 − ⎟ ; ⎝ Tc ⎠
( 0 ≤ t ≤ Tc )
447
[5.274]
Tc denotes the closing time of the gate. We are interested in determining the plane acoustic waves triggered in the pipe by the transient, assuming that there is a pressure node at the connection with the water tank.
Figure 5.34. Pipe provided with a sluice gate at one end and connected to a large water tank at the other
Before solving the problem as an application of the K. H. integral [5.269], it is useful to work out first the solution in a more standard way, that is by using directly the differential formulation of the problem. Actually, the method is essentially the same as that already presented in [AXI 05] to solve a longitudinal impact problem of a straight beam, so we shall not dwell here on the details but rather outline briefly the main steps and give the major results. The problem is first formulated as: ρ Q ∂ 2p 1 − 2 p = − f 0 ( U ( t ) − U ( t − Tc 2 ∂x cf S f Tc ∂p p ( x;0 ) = ( x;0 ) = 0 ∂t
Which is equivalent to:
;
)) δ ( L − x )
p ( 0; t ) = 0
[5.275]
448
Fluid-structure interaction
∂ 2p 1 − p=0 ∂ x 2 c 2f ρ Q ∂p ( L; t ) = + f 0 ( U ( t ) − U ( t − Tc ) ) ∂x S f Tc ∂p ( x;0 ) = 0 ; ∂t
p ( x;0 ) =
[5.276]
p ( 0; t ) = 0
Then, the Laplace transform is used to replace the partial derivative equation [5.276] by an ordinary differential equation, which can be easily solved to produce the Laplace transform of the pressure field in a compact analytical form as: p ( x; s ) = Z Z=
ρ f cf
Q0 ( e sΔτ − e − sΔτ ) (1 − e − sTc ) Tc s
;
Sf
τ = L / cf
2
∞
∑ ( −1) e
n − s (1+ 2 n )τ
n =0
;
[5.277]
Δτ = x / c f
Reverting to the time domain, by virtue of the delay theorem, the formula [5.277] is back transformed into the following series of retarded waves: p ( x; t ) =
-ρ Q t Z Q0 t Σ ( t ) − Σ ( t − Tc ) ) ; q ( x; t ) = f 0 ( Σ ( t ) − Σ ( t − Tc ) ) ( Tc Tc
[5.278]
Σ ( t ) stands for the following series of retarded Heaviside step functions: ∞
{
}
Σ ( t ) = ∑ ( −1) U ⎡⎣t + Δτ − (1 + 2n )τ ⎤⎦ + U ⎡⎣t − Δτ − (1 + 2n )τ ⎤⎦ n =0
n
[5.279]
To illustrate the features of such waves, let us consider first a slow transient Tc > 6τ and let us build the wave during a time interval 0 ≤ t ≤ 6τ , so that the
contribution of the retarded series Σ ( t − Tc ) vanishes. The pressure wave at tube end x = L can be determined by considering successive time intervals of duration 2τ which stands for the forth and back travel time of the waves. During the first interval 0 ≤ t ≤ 2τ , n = 0 and only the incident wave exists, which is: p0 ( x; t ) =
Z Q0t U (t ) Tc
[5.280]
During the next interval 2τ ≤ t ≤ 4τ , two waves occur, namely the n = 0 wave: p0 ( x; t ) =
Z Q0t ( U ( t ) − U ( t − 2τ ) ) = 2ZTQ0τ Tc c
[5.281]
3D Sound waves
449
and the n = 1 wave: Z Q0 ( 2τ − t ) -Z Q0 t U ( t − 2τ ) = Tc Tc
p1 ( L; t ) =
[5.282]
Therefore, the resulting wave is found to be: p ( L; t ) =
Z Q0 ( 4τ − t )
[5.283]
Tc
During the next interval 4τ ≤ t ≤ 6τ , three waves take place, whose expressions are found to be: n = 0 wave: p0 ( x; t ) =
Z Q0t 2Z Q0τ U ( t ) − U ( t − 2τ ) ) = ( Tc Tc
[5.284]
n = 1 wave: p1 ( L; t ) =
-Z Q0t ( U ( t − 2τ ) − U ( t − 4τ ) ) = -2ZTQ0τ Tc c
[5.285]
n = 2 wave: p2 ( L; t ) =
Z Q0t U ( t − 4τ ) Tc
[5.286]
The resulting wave is: p ( L; t ) = p2 ( L; t ) =
Z Q0t U ( t − 4τ ) Tc
[5.287]
Maximum pressure exerted on the gate is pmax =
2 Lρ f Q0 S f Tc
.
The case of a short transient Tc < τ can be treated in the same way by adding to the preceding wave the same wave shifted back in time by the quantity t − Tc . A numerical application is helpful to realise that in practice the pressure force can be quite substantial. A maximum pressure of two bars is reached, that is a force of 200 kN per square meters of gate, if the the following figures are assumed: L = 10 m ; ρ f = 1000 kg m −3 ; c f = 1000 ms −1 ; T = 1.0 s ; V0 = 10 ms −1 ρ f Q f = ρ f S f V0 = 100 kgs−1 ; pmax =
2.103 = 2.105 Pa = 2 bar 10−2
450
Fluid-structure interaction
Figure 5.35. Time profiles of the pressure wave
In Figure 5.35, the time profiles of the pressure wave are plotted, for a few positions along the tube x = L, x = L/2, x = L/4. All these curves have been calculated using the same procedure as that presented just above for the x = L case. As an exercise left to the reader, it is also interesting to plot the profiles of the volume velocity wave. The fluctuating volume velocity decreases linearly with time up to t = Tc , time at which its value is −Q0 precisely, as should be in order to cancel the steady outflow present initially. On the other hand, amplitude of the pressure wave increases as the closing time of the gate is shortened, as illustrated in Figure 5.36, which also brings in evidence the presence of violent peaks of negative pressure (i.e. tension) indicating that in reality cavitation is very likely to occur ( cf. Chapter 1 subsection 1.2.2.6). The kind of waves just described are responsible for the acoustic part of the so-called water hammer phenomenon which can be observed in various domestic and industrial pipe systems, when a tape or a gate is actuated in a too abrupt way. Pipe response to such transients will be addressed in Chapter 6. If now the integral solution [5.269] is adopted, the pressure field is obtained as the compact analytical form: ⎛ L − x ⎞ Z Qo 2 p ( x; t ) = p ⎜ L; t − ⎟+ ⎜ c f ⎟⎠ Tc ⎝
⎛ L− x ⎜⎜ t − cf ⎝
t−
x
⌠ cf ⎞ ⎮ ⎟⎟ − Z S f ⎮⎮ ⎮ ⎠ ⌡0
∂p ( 0; t0 ) dt0 ∂ x0
[5.288]
3D Sound waves
451
Figure 5.36. Time profiles of the pressure wave for a short transient at the end of the pipe, assuming there is no cavitation in the column of liquid
As in the case of formula [5.270], the remaining integral can be evaluated by writting the preceding result at the tube ends: ⌠
t
⎮ ∂ p Z Qo 2 p ( 0; t ) = p ( L; t − τ ) + ( t − τ ) − Z S f ⎮⎮ ( 0; t0 ) dt0 Tc ⎮ ∂ x0
[5.289]
⌡0
⌠
⎮ Z Qo t 2 p ( L; t ) = p ( L; t ) + −Z Sf ⎮ ⎮ Tc ⎮
t −τ
⌡0
∂p ( 0, t0 ) dt0 ∂ x0
[5.290]
Hence if t ≤ τ , we obtain: p ( L; t ) =
Z Qo t Tc
[5.291]
In the interval τ < t ≤ Tc , the integral is: t
⌠ ⎮ ∂ p cf ⎮ ⎮ ∂ x 0 ⎮ ⌡0
⎧
( 0; t0 ) dt0 = ⎨ p ( L; t ) + ⎩
Z Qo t ⎫ ⎬ U (t − τ ) Tc ⎭
[5.292]
452
Fluid-structure interaction t−
x
⌠ cf ⎮ cf ⎮ ⎮ ⎮ ⌡0
⎧ ZQ t ⎫ ∂p ( 0; t0 ) dt0 = ⎨ p ( L; t ) + o ⎬ U ∂ x0 Tc ⎭ ⎩
⎛ L+ x ⎞ ⎜⎜ t − ⎟ c f ⎟⎠ ⎝
[5.293]
Hence, in particular: Z Qo t Z Qo ( t − 2τ ) − − p ( L; ( t − 2τ ) ) Tc Tc
p ( L; t ) =
[5.294]
and: p ( x; t ) =
Z Qo Tc
⎛ L − x ⎞ Z Qo ⎜⎜ t − ⎟− c f ⎟⎠ Tc ⎝
⎛ L+ x ⎞ ⎜⎜ t − ⎟+ c f ⎟⎠ ⎝
[5.295]
⎛ ⎛ L − x ⎞⎞ ⎛ ⎛ L + x ⎞⎞ p ⎜ L; ⎜ t − ⎟⎟ ⎟ − p ⎜ L; ⎜⎜ t − ⎟⎟ ⎜ ⎜ ⎜ cf ⎠⎟ c f ⎟⎠ ⎟ ⎝ ⎝ ⎠ ⎝ ⎝ ⎠
In the above expressions, each individual wave component is made explicit and it is tacitly assumed that when the retarded times between parentheses take on negative values the corresponding term vanishes. Determination of pressure at a given position, x = L/2 for instance, follows as: p ( L / 2; t ) = 0≤t≤
τ : 2
Z Qo Tc
⎛ τ ⎞ Z Qo ⎜t − 2 ⎟ − T ⎝ ⎠ c
⎛ 3τ ⎜t − 2 ⎝
⎛ ⎛ 3τ ⎞ ⎟ − p ⎜ L; ⎜ t − 2 ⎠ ⎝ ⎝
⎞⎞ ⎟⎟ ⎠⎠
[5.296]
p( L / 2; t ) = 0
τ 3τ ≤t≤ : 2 2 3τ 5τ ≤t≤ : 2 2
p( L / 2; t ) =
Z Q0t Tc
p( L / 2; t ) =
Z Qo Tc
3τ ⎛ τ ⎜t − 2 −t + 2 ⎝
3τ ⎞ ⎛ ⎟ − p ⎜ L; t − 2 ⎠ ⎝
⎞ Z Qo ⎟ = T etc. ⎠ c
It is thus verified that this solution is equivalent to the series of retarded waves produced by the differential method. 5.3.5.4 K.H. integral for 3D external and internal problems
af
Let us consider once more a volume of fluid V either contained within a closed surface (S ) or lying between (S ) and another (fictitious) surface (S∞ ) , assumed to be located at an infinitely large distance from
(S ) .
We are interested here to
particularize the K.H integral [5.261] successively in the three following domains of
3D Sound waves
453
a receptor point (P). First (P) is assumed to lie outside the domain bounded by (S ) which is termed the external problem. Then, (P) is assumed to lie within the domain bounded by (S ) , which is termed the internal problem, and finally (P) is assumed to lie on the surface (S ) , which is termed the boundary surface problem. 1. The external problem
Figure 5.37. External problem
Since G decreases as 1 / r , the contribution of eventual sources laying on (S∞ ) is anyway zero. Thus the K.H. integral is written as: t
⌠ ⎮ ⎮ ⎮ ⎮ ⌡(V )
δ (r ) p ( r0 ; t0 ) δ ( t − t0 ) dt0 dr0 = 2 4π r
t
⌠ ⎮ ⎮ ⎮ ⎮ ⌡(S )
1 ⎡ p ( r0 ; t0 ) gradG.n ( r0 ) − G grad p ( r0 ; t0 ) .n ( r0 ) ⎤ d S dt0 ⎣ ⎦ ρf
⌠ ⎮ ⎮ ⎮ ⌡0 ⌠ ⎮ ⎮ ⎮ ⎮ ⌡0
[5.297]
where r0 designates the position vector of a current point of (S ) and n ( r0 ) the unit
vector normal to (S ) , pointing outside from (V ) , see Figure 5.37. Substituting the outgoing Green function [5.202] into [5.297] yields:
454
Fluid-structure interaction t
1 p (r;t) = 4π t
−
1 4π
⌠ ⎮ ⎮ ⎮ ⎮ ⎮ ⌡0
⌠ ⎮ ⎮ ⎮ ⎮ ⎮ ⌡0
⌠ ⎮ ⎧ ⎮ ⎪ ⎮ ⎨p ⎮ ⎮ ⎪ ⎩ ⎮ ⌡(S )
⌠ ⎮ ⎧ ⎮ ⎪1 ⎮ ⎨ δ ⎮ ⎪r ⎮ ⎩ ⌡(S )
⎛ ⎡ 1 ⎛ r δ ⎜ t − t0 − ⎜ r c ⎢⎣ f ⎝ ⎝
( r0 ; t0 ) ⎜⎜ grad ⎢
⎛ r ⎜⎜ t − t0 − cf ⎝
⎞ ⎤ ⎞ ⎫⎪ ⎟⎟ ⎥ ⎟ .n ( r0 ) ⎬dt0 dS ⎪⎭ ⎠ ⎥⎦ ⎟⎠
[5.298] ⎞ ⎫⎪ ⎟⎟ grad p ( r0 ; t0 ) .n ( r0 ) ⎬dt0 dS ⎪⎭ ⎠
(
)
or, in the spectral domain: ⌠
iω r ⎮ ⎧ ⎛ ⎡ 1 − iωc r ⎤ ⎞ ⎮ ⎪ 1 − cf f ⎜ ⎟ ⎮ 4π p ( r ; ω ) = ⎥ − grad [ p( r0 ; ω )] e ⎨ p( r0 ; ω ) grad ⎢ e ⎮ ⎜ r ⎢⎣ r ⎥⎦ ⎟⎠ ⎪ ⎮ ⎝ ⎩ ⎮
(
⌡(S )
)
⎫ ⎪ ⎬ .n ( r0 )dS ⎭⎪
[5.299] 2. The internal problem The calculating process is the same as for the external problem. So it leads to the same formulas as [5.298] and [5.299], except that the sign is changed because we use the ingoing Green function instead of the outgoing one and the normal unit vector n is pointing toward the interior of volume (V ) . Such a result indicates that the pressure field undergoes a finite discontinuity when crossing the boundary surface (S ) . 3. The boundary surface problem It can be also needed to calculate the pressure field at the surface itself (S ) . In contrast with either the external or the internal problems, the Green function becomes singular in the integration domain which includes the value r = 0 . Such a difficulty can be avoided by modifying slightly the integration path in such a manner as to remove the singular point from the integration domain. For that purpose, the receptor point is surrounded by a hemispherical cap of infinitesimal radius ε, denoted (Sc ) , closed by the complementary part (Sε ) of the boundary
(S ) , see Figure 5.38.
3D Sound waves
455
Figure 5.38. Integration path avoiding the singular point P
Then the external problem is integrated over (Sc ) and (S ) ∩ (Sε ) . As (P) is removed from the integration domain, the K.H. integral comprises surface terms, only. Using the spectral domain, equation [5.299] becomes:
0=
⌠ ⎮ ⎮ ⎮ ⎮ ⌡(Sc )
⌠ ⎮ − ⎮⎮ ⎮ ⌡(S -Sε )
1 e r
iω r − cf
iω r
1 − cf e r
⌠
(
⎮ ⎮ grad p .ndS dt + ⎮
)
S dt ( grad p ) .nd
⎮ ⎮ ⌡(S -Sε )
⌠ ⎮ ⎮ +⎮ ⎮ ⎮ ⌡(Sc )
iω r ⎛⎜ ⎛ 1 − c f p grad ⎜ e ⎜r ⎜ ⎝ ⎝
⎛ ⎛ 1 p ⎜ grad ⎜ e ⎜r ⎜ ⎝ ⎝
⎞⎞ ⎟ ⎟ .ndS dt + ⎟⎟ ⎠⎠
[5.300] iω r − cf
⎞⎞ ⎟ ⎟ .ndS dt ⎟⎟ ⎠⎠
As ε tends to zero, it is found that the integral over the hemispherical cap can be written as: ⌠ ⎮ ⎮ ⎮ ⎮ ⎮ ⌡(Sc )
⎛ ⎛ 1 − c p ⎜ grad ⎜ e f ⎜r ⎜ ⎝ ⎝
⌠ ⎮ ⎮ ⎮ ⎮ ⌡(Sc )
1 − cf e r
iω r
iω r
iωε ⎞⎞ ⎛ ⎛ 1 iω ⎞ − c ⎟ ⎟ .nd S −2πε 2 ⎜ p( r ; ω ) ⎜ 2 + ⎟ e f ⎟⎟ ⎜ ε ⎠ ⎝ε ⎠⎠ ⎝
Σ 2πε ( grad p ) .nd
2
iωε ⎛ ⎛1⎞ −c ⎜ p( r , ω ) ⎜ ⎟ e f ⎜ ⎝ε ⎠ ⎝
⎞ ⎟ −2π p( r ; ω ) ⎟ ⎠
⎞ ⎟0 ⎟ ⎠
Therefore, letting ε tend to zero, the following pressure field is obtained:
[5.301]
456
Fluid-structure interaction ⌠ ⎮
⎮ 2π p ( r ; ω ) = ⎮
⎮ ⎮ ⌡(S )
⎛ ⎛ 1 − c p ⎜ grad ⎜ e f ⎜r ⎜ ⎝ ⎝
iω r
⌠
iω r ⎞⎞ ⎮ 1 − c f ⎮ ⎟ ⎟ .ndS − ⎮ e grad p.ndS ⎟⎟ r ⎮ ⎠⎠ ⌡(S )
[5.302]
The expression [5.302] differs from [5.299] by the multiplication factor two solely, in full agreement with the analysis of sound radiation in the presence of perfectly reflecting plane boundaries based on the image method, cf. subsections 5.4.3.2. and 5.4.3.3. The K.H. integrals form the basis of computational techniques known as the boundary element methods (BEM), which are of particular interest to deal with large enclosure and open-space problems. The basic idea of the method is to discretize the K.H. integrals based on a mesh of the boundary surfaces, defined typically as a collection of quadrangular panels, or facets, where pressure and its normal derivative is assumed to be constant. As already mentioned in Chapter 2, this topical subject remains however far beyond the scope of the present book, mainly because several numerical aspects are difficult to master without entering into arduous technical details, which are beyond the purview of the authors. The reader interested in this domain, which is still largely open to research and motivate an abundant literature, can be referred to [BRE 78], [HEC 92] for an introductory overview and references to the specialized literature. To conclude the present subsection, the following remarks may be useful. NOTE 1.–
BEM versus FEM
As clearly shown by the K.H. integral formulation, to model an acoustical problem by using the boundary element method, it is sufficient to mesh the boundary surface (S ) of the fluid domain and to define both the pressure field p and the normal gradient field grad p.n on it, as external sources. The obvious advantage in comparison with the finite element method is to avoid the necessity to mesh large volumes by using 3D finite elements, reducing thus the number of degrees of freedom by an order of magnitude, at least. Such an advantage is of considerable practical importance, and is even decisive, in handling many problems involving large, and even infinite volumes of fluids. However, as the method is basically based on the analytical form of a Green function, severe difficulties arise to deal with dispersive and nonlinear problems. NOTE 2.–
Nature of the surface sources
The pressure field used as a surface source appears in the K.H. integral as a dipole term. Physical interpretation of it is perhaps made more intuitive if the pressure force on (S ) is considered. On one hand, it depends on the sound waves
3D Sound waves
457
impinging on and being reflected (and eventually absorbed) by (S ) ; on the other hand it also depends on the normal motion of (S ) . The normal gradient field is proportional to the normal acceleration of the fluid at (S ) . Therefore, used as a surface source, it is interpreted as a volume velocity source, which appears in the K.H. integral as a monopole term. On the other hand, it should be emphasized once more that the pressure field and the normal acceleration distributions on (S ) cannot be independent, as they are related to each other by the impedance of the boundary. As already mentioned at the end of subsection 5.3.5.1 the pressure on (S ) is a dependent variable field, which appears both on the left and the right-hand sides of the K.H. integral equations. As a particularily simple and often encountered case, part of the boundary can be assumed to be fixed and the complementary part oscillating. In that case, grad p.n is known everywhere and pressure is determined on the basis of the duplication property, as already shown in the case of plane waves in subsection 5.3.5.2 and 5.3.5.3. Once the surface fields p and grad p.n are known everywhere on (S ) they can be computed for any field points outside (S ) . If pressure can be assumed to be zero on a part of the boundary, the duplication property is used to determine grad p.n , as further explained on a specific example in subsection 5.3.5.5. NOTE 3.–
Low frequency range
In the low frequency range, fluid compressibility becomes unimportant and the K.H. integral can be simplified accordingly. Considering the spectral version of it, equation [5.299] simplifies into the incompressible formula: ⌠
⎮ 4π p( r ; ω ) = ⎮⎮
⎮ ⌡( Σ )
⎧ 1⎫ ⎛ ⎛ 1 ⎞ ⎞ ⎨ p( r0 ; ω ) ⎜ grad ⎜ ⎟ ⎟ − grad ( p( r0 ; ω ) ) ⎬ .n ( r0 )d S r⎭ ⎝ r ⎠⎠ ⎝ ⎩
(
)
[5.303]
An application of this limit case will be presented in Chapter 7, subsection 7.2.1.3. 5.3.5.5 Application: pressure field induced by the unbaffled circular piston The problem of the pressure field induced in a fully open medium by a circular disk vibrating perpendiculary to its own plane is shown in Figure 5.39. As already stated in subsection 5.3.4.9 based on qualitative reasoning, a device of this kind is much less efficient as a sound radiator than the baffled version of it. Here a quantitative analysis of the problem is made, based on the K.H. integral. For the
458
Fluid-structure interaction
sake of mathematical simplicity it is restricted to the far field approximation, which by the way, is suitable for handling the problem of sound radiation. So in such a problem, the physical boundary is restricted to the two opposite faces of the disk, assumed to be of negligible thickness, and it is useful to consider the portion of the plane z = 0 outside of the disk as a fictitious boundary which divides the medium into two halfspaces provided with the same response properties. As, on the other hand, both faces of the disk excite the fluid also in exactly the same way, the problem is clearly antisymmetric about z = 0, which implies the following conditions, written first at the disk surface and then on the fictitious boundary, drawn as a dashed line in Figure 5.39.
Figure 5.39. Problem of the unbaffled circular piston
In the range z = 0; r ≤ R0 , as defined in cylindrical coordinates, the acceleration field is necessarily the same on both faces of the disk. Therefore the following relation holds concerning the normal pressure gradient: ∂p+ ∂z
= z =0+
∂p− ∂z
r < R0
[5.304]
z =0−
On the other hand, as the fluid on one face is pushed at the same time as the fluid is pulled on the other face, the following relation holds concerning the pressure: p+
z =0
= − p−
z =0
= p0
r < R0
[5.305]
In the range outside the disk z = 0; r > R0 , there is no pressure source, hence the pressure field must be continuous accross the fictitious boundary. Thus the following relation holds concerning the pressure: p+
z =0
= − p−
z =0
= 0 r ≥ R0
[5.306]
3D Sound waves
459
The normal gradient is still governed by the conditon of antisymmetry, which states that the volume velocity along the n+ direction is equal and opposite to that along the n− direction: ∂p+ ∂z
= z =0+
∂p− ∂z
r ≥ R0
[5.307]
z =0−
To carry out the integration on the two faces of the z = 0 plane (i.e. the actual plus the fictitious boundary), it suffices to sum the terms subscripted by the plus sign to those subscripted by the minus sign. In the process, the contribution of the monopole source cancels out by virtue of the conditions [5.305] and [5.306]. Therefore the K.H. integral reduces to the dipole component and the domain of integration is restricted to the the disk surface denoted (Sd ) : ⌠
t ⎮ ⌠2 p( r ; t ) = ⎮⎮ dt0 ⎮⎮
⌡t1
⎮ ⌡(Sd )
⌠
t ⎮ ⌠2 1 ( p+ n+ + p- n- ) .grad GdS = 2⎮⎮ dt0 ⎮⎮ ρf ⌡t1 ⎮
⌡(Sd )
1 ∂G p0 dS ρf ∂z
[5.308]
Or shifting to the spectral domain, described here in terms of wave numbers as: ⌠
⎮ p( r ; k ) = ⎮⎮
⎮ ⎮ ⌡(Sd )
∂G 2 p0 ∂z0
dS = z0 = 0
⌠ ⎮ ⎮ 2 p0 ⎮ ⎮ ⎮ ⌡(Sd )
∂ ⎛ e − ikr ⎞ ⎜ ⎟ dS ∂z0 ⎝ 4π r ⎠ z =0 0
[5.309]
Approximate computation of the integral [5.309] proceeds along the same lines as those described in subsection 5.3.4.8. However, due to the dipolar nature of the source term in [5.309], the expression used to approximate the distance r between a far field receptor point and a current emissive point must include explicitely the z dependency of the source position in order to carry out the differentiation included in formula [5.309]. After a few manipulations, the following approximation is retained, which extends the relation [5.246] to the 3D case: r r − r0 sin ϕ cos θ − z0 cos ϕ The derivative of G is written as: ⎛ e − ik ( r − r0 sin ϕ cosθ ) ⎞ ∂ ⎛ ⎞ ∂G eikz0 cosϕ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ∂z0 z = 0 ⎝ 4π ⎠ ∂z0 ⎝ r − r0 sin ϕ cos θ − z0 cos ϕ ⎠ z0 = 0 0
The result approximated to the first order terms in 1/r is found to be:
[5.310]
[5.311]
460
∂G ∂z0
Fluid-structure interaction
z0 = 0
⎛ ik cos ϕ ⎞ − ik ( r − r0 sin ϕ cosθ ) ⎜ ⎟e ⎝ 4π r ⎠
[5.312]
In principle, to proceed to the solution of the integral equation [5.309], it would be necessary to substitute the weighting function [5.312] and to determine the pressure field at the surface of the disk. However, to understand the major features of the radiated sound field, the problem can be simplified by assuming that the unknown pressure p0 is distributed uniformly on the faces of the disk and its value is taken as a free parameter, proportional to the fluid acceleration. Exact calculation is postponed to Chapter 6. Designating by F the magnitude of the resulting pressure force exerted on the disk, the integral [5.309] is written in the simplified form: 2π ⎛ F ⎞ ⎛ ik cos ϕ e − ikr ⎞ ⌠ R0 ⌠ − ikr sin ϕ cos θ ⎮ r0 dr0 ⎮ e 0 p( r ; k ) = ⎜ dθ ⎟ ⎜ ⎟ ⎮ 2 ⎮ ⌡0 4π r ⎠ ⌡0 ⎝ π R0 ⎠ ⎝
[5.313]
As the remaining integrals are identical to those already encountered in subsection 5.3.4.8, see equation [5.247], the pressure field [5.313] can be put directly in its final form, suitable for a comparison with the baffled case: ⎛ ikFe − ikr p( r, ϕ ; k ) = ⎜ ⎝ 4π r
⎞ ⎛ 2 J 1 ( kR0 sin ϕ ) ⎞ ⎟ cos ϕ ⎟⎜ ⎠ ⎝ kR0 sin ϕ ⎠
[5.314]
In full agreement with the qualitative analysis of subsection 5.3.4.9, if the frequency tends to zero, the radiated pressure tends also to zero independently of the dipole strength F. On the other hand, as in the case of the baffled piston, radiation is found to be directive. Independently of the frequency or wavelength considered, the magnitude of the pressure is maximum along the dipole axis and nil in the plane of the disk, as should be. The directivity factor within the parentheses is the same as for the baffled piston.
Chapter 6
Vibroacoustic coupling
An introductive description of the coupling mechanism between the vibration of structures and the wave motion in a fluid has been already presented in Chapter 3, in the context of gravity waves. Coupling between sound waves and structural vibrations gives rise to the so-called vibroacoustic waves which are of practical importance in many fields concerning acoustic and structural engineering. They are analysed in the present chapter in some depth, restricting however the problem to the conservative case, which means in particular that both the solid and the fluid are assumed to be of finite extent, and limited by perfectly reflecting boundaries. Presentation is first focused on one-dimensional systems, which are amenable to analytical and semi-analytical solutions based on the modal synthesis method. The coupled problems will be formulated in terms of non symmetrical as well as symmetrical equations, depending on the field variables used to describe the fluid. In this respect, several examples are worked out to highlight interesting physical features of vibroacoustic coupling and illustrate a few numerical aspects of practical relevance concerning the computed vibroacoustic modes. As a short and particularly interesting incursion into the nonlinear domain, the vibroacoustic response of a pipe to a cavitating liquid is analysed based on a simplified model of cavitation which emphasises the marked similarities of the phenomenon with both plasticity and impacts problems. The chapter is concluded by establishing the variational formulations which are useful to build a finite element model of the fluid-structure coupled system. Of particular interest is the mixed and symmetric formulation, in which the fluid is described by using two variables, namely the pressure and another quantity closely related to the displacement potential of the fluid. It turns out that the method presents very significant advantages versus non symmetrical formulations in terms of convenience in programming and computational efficiency.
462
Fluid-structure interaction
6.1. Local equilibrium equations 6.1.1
Mixed and non symmetrical formulation
The dynamical equilibrium of a fluid-structure coupled system is described by using the equations [2.1] or [2.4] except that here fluid compressibility is taken into account as in linear acoustics. Considering harmonic oscillations, the system [2.4] becomes: K s ⎡⎣ X s ⎤⎦ − ω 2 M s ⎡⎣ X s ⎤⎦ + pnδ ( r − r0 ) = 0 2
⎛ω ⎞ Δp + ⎜ ⎟ p − ω 2 ρ f X s .nδ ( r − r0 ) = 0 ⎝ ce ⎠ + (C.B.C)
[6.1]
Again (C.B.C) is an abbreviation for “Conservative Boundary Conditions” and r0 is the position vector of a current point at the fluid-structure interface, that is the wetted wall (W ) . The mean properties of the fluid are the density ρ f and the
effective speed of sound ce which takes into account the change of the fluid volume induced by the dilatation of the walls. The system [6.1] is also written in terms of ordinary functions as: K s ⎡⎣ X s ⎤⎦ − ω 2 M s ⎡⎣ X s ⎤⎦ = 0 2
⎛ω ⎞ Δp + ⎜ ⎟ p = 0 ⎝ ce ⎠ σ s .n (W ) = − pn (W )
; grad p.n
(W )
= ω ρ f X s .n
[6.2]
2
(W )
+ (C.B.C.)
We recall that σ s is the local stress tensor of the solid. As in the incompressible case, the motion of the fluid can be entirely described in terms of the pressure field. However, it is important to emphasize that the physics described by the modal equations [6.1] or [6.2] differ profoundly from that which is governed by the modal equations [2.1] or [2.4] because now the fluid behaves as an elastic continuum. Hence, it provides the coupled system with its own infinitely many degrees of freedom. So there are infinitely many natural modes of vibration of the coupled system which are described in terms of structural displacement and fluid pressure. On the other hand, the vibration of the coupled system can be analysed either by solving directly the system [6.2] or by projecting it first onto a modal basis, which comprises a suitable set of structural modes of vibration and acoustical modes of the fluid. As these modes are coupled by the conditions to be fulfilled at the wetted wall (W ) , the physical mechanism is termed vibroacoustic coupling and the coupled modes are termed vibroacoustic modes.
Vibroacoustic coupling
6.1.2
463
Symmetrical formulation in terms of displacements
As already pointed out in Chapters 1 and 2, the vibration of the fluid can also be described by using a displacement field X f , leading to the following system of equations:
K s ⎡⎣ X s ⎤⎦ − ω 2 M s ⎡⎣ X s ⎤⎦ = 0
curl X f = 0 ; σ s .n
(W )
= + E f divX f n
−grad E f divX f − ω 2 ρ f X f = 0 −grad E f divX f .n (W ) = ω 2 ρ f X s .n
(
(W )
[6.3]
)
(
)
(W )
where E f = ρ f ce2 . The second line of system [6.3] states that fluid motion is restricted to the class of potential flows. The condition is automatically fulfilled in 1D problems, but not in 2D or 3D geometries. It is worth noticing that constraining fluid motion to be irrotational leads to computational difficulties inherent in the displacement formalism, except of course in 1D problems. Furthermore, as particular solutions of the modal system [6.3], we find the steady and incompressible fluid flows about the fixed structures which comply with the conditions ω = 0 ; divX f = 0 ; X s = 0 . Such solutions are clearly of no interest so far as fluid-structure coupled problems are concerned. It may be useful to point out, for the sake of clarity at least, that line four of system [6.3] simply states that the normal stress exerted on the solid at the wall is equal to the pressure in the fluid, changed of sign because, as a convention in structural mechanics tension is a positive and pressure a negative stress. Furthermore, pressure is here given by the elastic law p = − E f divX f .n . (W )
On the other hand, though it is generally preferred to use the mixed equations [6.1] rather than [6.3], the latter have the theoretical advantage to treat mathematically the solid and the fluid on the same basis, except the specificities inherent in the elastic laws which differ in both media. In particular, if the formalism [6.3] is selected to describe the fluid-structure system, the concepts and analytical techniques of the modal expansion methods described in [AXI 05] in the context of structural mechanics can be directly applied to solve the vibroacoustic problem. For such a purpose, it is appropriate to replace the fluid-structure coupling terms expressed in [6.3] as interfacial stresses by the normal contact condition [1.59], repeated here for convenience: X f − X s .n = 0 ⇔ X f − X s .n = 0 ⇔ X f − X s .n =0 [6.4]
(
)
(W )
(
)
(W )
(
)
(W )
With the aid of the contact condition [6.4], the system of equations [6.3] is transformed into the following one:
464
Fluid-structure interaction
K s ⎡⎣ X s ⎤⎦ − ω 2 M s ⎡⎣ X s ⎤⎦ + K L X s − X f .nδ ( r − r0 ) = 0 − grad E f divX f − ω 2 ρ f X f + K L X f − X s .nδ ( r − r0 ) = 0
(
)
(
(
)
)
[6.5]
The form of [6.5], which is obviously symmetrical, shows that the structure and the fluid are treated on the same basis as two distinct mechanical subsystems which are rigidly connected at the interface. Accordingly, the stiffness coefficient K L is used as a penalty factor to prescribe the desired condition X s .n = X f .n at the fluidstructure interface. In an equivalent way, it can be interpreted as a Lagrange multiplier related to the holonomic condition [6.4]. In the remainder of this chapter, the formalism [6.5] will be used in parallel with equations [6.1] to treat several onedimensional problems selected as illustrative examples. 6.1.3
Mixed and symmetrical formulation
To implement the fluid-structure inertial and vibroacoustic coupling mechanisms in finite element computer programs, it is advantageous to reformulate the coupled equations [6.1] in an equivalent form which complies with the conservative nature of the problem, as the solid part of the problem does. Hence, a self-adjoint formulation equivalent to [6.1] is required. It turns out that such a transformation is possible. The key point is to introduce an additional variable which describes the fluid together with the pressure field. The method was first proposed by Ohayon and Valid, see [MOR 79], [OHA 81] and also [MOR 95], and then implemented and validated in a large purpose finite element computer program by Gibert and Axisa, see [GIB 78]. Hereafter, following the notations used in [GIB 78], [AXI 82] and CASTEM2000 software, the new variable is denoted Π . As a definition it is related to the pressure by: p = Π ⇔ p = −ω 2 Π [6.6] The physical meaning of Π can be made clear using the momentum equation: 1 1 1 X f + [6.7] grad p = 0 ⇔ X f = − gradΠ ⇔ X f = − gradΠ ρf ρf ρf The last equality in [6.7] holds because the fluctuating quantities X f and Π are centred, that is their mean value is zero, as long as we deal with oscillatory motions about a state of equilibrium at rest. On the other hand X f , hence Π , can be expressed in terms of the displacement potential Φ : X f = −gradΦ ⇒ Π = ρ f Φ
[6.8]
After a few elementary manipulations, the system of equations [6.1] is transformed into the symmetrical form:
Vibroacoustic coupling
465
K s ⎡⎣ X s ⎤⎦ − ω 2 M s ⎡⎣ X s ⎤⎦ + ω 2 Π nδ ( r − r0 ) = 0 ⎛ 1 p ⎞ −ω 2 ⎜ ΔΠ − + ω 2 X s .nδ ( r − r0 ) = 0 2 ⎟ ⎜ρ ⎟ ρ f cf ⎠ ⎝ f
[6.9]
With the aid of Π , one is able to include all the fluid-structure interaction terms into a single vibroacoustic operator, which is similar to a mass operator in the specific meaning that it operates on the second time derivatives of the solid and fluid variables. On the other hand, since the p and Π variables used to describe the fluid are not independent from each other, some mathematical peculiarities arise, namely the mathematical occurrence of infinitely many eigenmodes at zero frequency, as further discussed in subsections 6.2.1.4 and 6.3.2.5 using illustrative examples. 6.2. Piston-fluid column system The simplest conceivable vibroacoustic system is shown in Figure 6.1, which actually was already introduced in Chapter 2 as a first elementary example, in the context of coupling mechanism by fluid inertia. Here the speed of sound in the fluid is assumed to have some finite value, denoted ce to emphasize that it can be lower than in infinite fluid, due to the dilatation of the walls of the tube (cf. Chapter 4, subsection 4.4.4). The right-hand side end of the tube, chosen conventionally as the tube outlet, is assumed to be terminated by the real dimensionless impedance α out .
Figure 6.1. Spring mass system coupled to a compressible fluid column in a rigid tube
The object of this section is to use this archetypical system to analyse the vibroacoustic natural modes and the forced response to various kinds of excitation. The vibroacoustic modal problem is solved first analytically to produce solutions in closed analytical form. They serve as a reference to discuss the numerical solutions produced by the modal projection method. The latter is described in some depth by considering successively three distinct formulations of the vibroacoustic coupling mechanism. Presentation of the forced problems will follow the same lines as the modal problem, by using first the analytical method which consists in solving the local equations of the type [6.1], using either a Fourier or a Laplace transform. The analytical formalism is well suited to point out the major effects brought by the
466
Fluid-structure interaction
vibroacoustic coupling mechanism to the dynamical response of the coupled system. Then, the modal projection method is worked out by considering successively the three distinct formulations introduced earlier to solve the modal problem. Finally, it is shown that the transfer matrix method, introduced in Chapter 4 to deal with plane wave acoustics in pipe and ducts, can be extended to treat one dimensional vibroacoustic problems in the spectral domain. 6.2.1
Modal problem
6.2.1.1 Analytical solution The system [6.1] is written as:
( K δ ( x ) − ω M δ ( x )) X 2
s
s
s
+ p ( x, ω ) S f δ ( x ) = 0
2
d2p ⎛ ω ⎞ + ⎜ ⎟ p − ω 2 ρ f X sδ ( x ) = 0 dx 2 ⎝ ce ⎠ Zout =
i ρ f ce Sf
[6.10]
tan α out
In this particular example, the solid is a discrete system (a single DOF) whereas the fluid is a continuum. Therefore, to unify the mathematical treatment of the solid and the fluid parts of the problem, the coefficients of the mass-spring system are multiplied by the Dirac distribution δ ( x ) , as already explained in Chapter 2, subsection 2.2.2.2. The pressure field induced by the harmonic vibration of the piston is found to be: ⎛ ⎛ ωx ⎞ ⎛ ωL ⎞ ⎛ ωx ⎞⎞ + αout ⎟ cos ⎜ p ( x; ω ) = ωρ f ce X s ⎜⎜ sin ⎜ ⎟ − tan ⎜ ⎟ ⎟⎟ ⎝ ce ⎠ ⎝ ce ⎠ ⎠ ⎝ ⎝ ce ⎠
[6.11]
Using [6.11], the equation governing the motion of the piston is found to be: ⎛ ⎛ωL ⎞⎞ ω ce 2 + αout ⎟ ⎟⎟ X s = 0 M f tan ⎜ ⎜⎜ K s − ω M s − L c ⎝ e ⎠⎠ ⎝
[6.12]
M f = ρ f S f L is the physical mass of the fluid contained in the tube. The infinitely
many roots of the transcendental equation [6.12] stand for the natural frequencies of the vibroacoustic modes of the system. To discuss the qualitative features of these modes in relation to the properties of the fluid and the structure, it is found convenient to rewrite [6.12] in the following dimensionless form: κ 02 − κ 2 = μ f κ tan (κ + α out )
[6.13]
where the following reduced quantities are used: ω02 = K s / M s
; μf = M f / Ms
; κ = ω L / ce = kL ; κ 0 = ω0 L / ce = k0 L [6.14]
Vibroacoustic coupling
467
The force exerted by the fluid on the piston can be written either as: Ff ω2M f X s
=
tan (κ + αout ) κ
[6.15]
or as: Ff L Ef Sf Xs
= κ tan (κ + αout )
[6.16]
In [6.15], the scaling factor used to reduce the fluid force is the inertia force of the oscillating fluid column, whereas the scaling factor used in [6.16] is the elastic force of the fluid column fixed at one end and free at the other. In fact, in the long wave range κ << 1 , F f reduces practically to a purely inertia force if the tube is open at the outlet (α out = nπ ) : ⎛ lim ⎜ ⎝
Ff
κ →0 ⎜ ω 2 M
⎞ ⎛ tan κ ⎞ ⎟⎟ = κlim ⎜ ⎟ =1 → 0 ⎝ κ ⎠ f Xs ⎠
[6.17]
The result [6.17] means that the fluid is moving practically as an incompressible column of mass M f at the same velocity as the piston. Furthermore, the vibroacoustic force μ f κ tan κ is found to be larger than the inertia force μ f κ 2 since κ tan κ is larger than κ 2 . Therefore, the frequency of the first vibroacoustic mode is found to be lower than the purely inertial value, which is not an intuitive result.
On the other hand, if the tube is closed at the outlet ( α out = ( n + 1/ 2 ) π ), the force exerted by the fluid on the piston is expected to comprise both elastic and inertia components, since fluid motion is not entirely prevented as long as fluid compressibility is not zero. This can be easily verified by expanding [6.15] or [6.16] to the second order in κ: ⎛ ⎞ F ⎛ − cot κ = lim lim ⎜ 2 f ⎟ κ →0 ⎜ ω M X ⎟ κ →0 ⎜⎝ κ f s ⎠ ⎝
⎞ ⎛ 1 1⎞ ⎟ −⎜ 2 − ⎟ 3⎠ ⎠ ⎝κ
[6.18]
It follows that in the long wave range κ << 1 , the force exerted by the fluid on the piston can be expressed as: Ff −
Ef Sf L
Xs +
ω2M f X s 3
= −K f X s +
ω2M f X s 3
[6.19]
As could be also anticipated, the quantities M f and K f = E f S f / L are suitable to scale the elastic and inertia components of F f , respectively. More generally, the fluid force is approximated to the second order in κ by one of the two following forms:
468
Fluid-structure interaction
⎛ 2κ + ( 2 − κ 2 ) tan αout ⎞ ⎟ Ff = ω M f X s ⎜ ⎜ κ ( 2 − κ 2 − 2κ tan αout ) ⎟ ⎝ ⎠ 2
(
2 E S X ⎛ κ 2κ + ( 2 − κ ) tan αout Ff = f f s ⎜ ⎜ ( 2 − κ 2 − 2κ tan αout ) L ⎝
) ⎞⎟
[6.20]
⎟ ⎠
Let us consider, for instance, the elastic impedance [4.57], repeated here for convenience as: tan α out =
− K out − K out L −γ out = = ωρ f ce S f κ E f S f κ
[6.21]
where γ out is defined as the ratio of the terminal stiffness coefficient to that of the fluid column. Substituting [6.15] into the last form of [6.14], the fluid force is found to be:
(
2 2 E f S f X s ⎛ 2κ − ( 2 − κ ) γ out ⎜ Ff = ⎜ L 2 − κ 2 + 2γ out ⎝
) ⎞⎟ ⎟ ⎠
[6.22]
As could be anticipated, the force is essentially inertial or elastic in nature depending on whether γ out is much smaller, or, much larger than one. However, for intermediate values of κ, the force described by [6.22] cannot be separated into distinct inertia and stiffness components, even in the long wave range, except if κ 2 approaches zero, in which case F f is approximated to as the purely elastic force: ⎛ K K F f = − ⎜ out f ⎜K +K f ⎝ out
⎞ ⎟⎟ X s ⎠
[6.23]
The result [6.23] merely means that the fluid column and the terminal impedance act as two springs mounted in series. Because of the presence of the factor tan (κ + α out ) in the fluid force [6.15] or [6.16], the asymptotic behaviour of the system is not so easy to grasp if κ becomes very large, or when it is very small, mathematically at least. It can be noticed that a large value of κ implies that the characteristic time of acoustic wave propagation through the tube is much larger than the period of vibration of the structure. Therefore, the fluid and the solid are likely to be poorly coupled to each other. In particular, it can be claimed that the roots of the transcendental equation cot (κ + α out ) = 0 correspond necessarily to the natural frequencies of the purely acoustical modes, since any motion of the piston at those frequencies would lead to a fluid force of infinite magnitude, which is clearly unphysical.
Vibroacoustic coupling
469
(a)
(b) Figure 6.2. Open tube, characteristic equation [6.7]. The roots are indicated by circles and stars
470
Fluid-structure interaction
(a)
(b) Figure 6.3. Closed tube: characteristic equation [6.7]. The roots are indicated by circles and stars
Vibroacoustic coupling
471
Finally, the intermediate range of κ values must be investigated by solving numerically the characteristic equation [6.7], as shown graphically in Figures 6.2 and 6.3 for κ ranging from zero to twelve. Figure 6.2 refers to the case of an open outlet. The natural frequency of the first vibroacoustic mode is shown in plot (a), denoted by small circles. It behaves as a decreasing function of μ f in qualitative agreement with the fluid force [6.17], according to which inertia increases with μ f , while stiffness coefficient is maintained constant. For instance, if μ f = 1 and κ o = 3 , the first four roots of the characteristic equations are found to be: κ1 1.378 ; κ 2 3.045 ; κ 3 5.0135 ; κ 4 7.998 .
The first value can be compared to the natural frequency of the mass-spring system coupled to the incompressible fluid, which is substantially higher: κˆ1 2.120 instead of κ 1 1.378 , in agreement with what was stated above. On the other hand, the frequencies of the acoustical resonances of the tube, assuming α in = π / 2 + nπ and α out = nπ , are: κ1 = π / 2 1.5708 ; κ 2 3π / 2 4.7123 ; κ 3 = 5π / 2 7.854 etc. As indicated by the star symbols in plot (b), if μ f is sufficiently small, the roots κ n tend precisely to those values as n increases. Such a trend can be easily understood because at high frequency the piston remains essentially at rest whereas the fluid vibrates according to a closed-open acoustical mode. Conversely, if μ f is large, as n increases, the roots κ n tend to the acoustical resonances of the tube terminated by an inertial impedance M s at the inlet and by a pressure node at the outlet. Figure 6.3 refers to the case of a tube closed at the outlet. In contrast with the open tube, the first vibroacoustic natural frequency is found to increase with μ f . On the other hand, as indicated by the star symbols in plot (b), if μ f is sufficiently small, the roots κ n tend to nπ as n increases, corresponding thus to the acoustical resonances of the fluid column closed at both ends. Such a result is a mere consequence of the fact that at high frequency the mass-spring system behaves as a fixed wall, because of its large inertia. 6.2.1.2 Modal expansion method: displacement as the fluid variable As already outlined in subsection 6.1.1.2, the general principles of the modal substructuring method are also well suited for solving the present problem by treating the fluid and the structure as two distinct elastic subsystems connected to each other at the fluid-structure interface. Here, to keep closely in line with the presentation made in the context of structures, the fluid is described in terms of the displacement field X f . Equations [6.4] are particularized here as:
472
Fluid-structure interaction
(( K
s
− ω 2 M s ) X s + KL ( X s − X f
−E f S f
∂2 X f ∂ x2
))δ ( x ) = 0 [6.24]
− ω 2ρ f S f X f + KL ( X f − X s )δ ( x ) = 0
The value of K L must be sufficiently large with respect to the generalized stiffness of the computed modes in order to verify the desired condition X s = X f to a sufficiently good degree of accuracy. To be more specific, the appropriate value of K L is the so-called stiffness coefficient of modal truncation, as defined in [AXI 05], Chapter 4. Before projecting equations [6.24] onto a suitable modal basis of each subsystem, it is found appropriate to rewrite [6.24] in a dimensionless form, using the scaling factors [6.14] in conjunction with the stiffness ratio: γL =
KL Ks
[6.25]
With the aid of a few elementary manipulations, the following reduced equations are obtained:
( (κ
2 0
) ) δ (ξ ) = 0
− κ 2 ) X s + γ Lκ 02 (X s − X f 2
⎛∂ Xf −μ f ⎜ + κ 2X f ⎜ ∂ξ 2 ⎝
⎞ 2 ⎟⎟ + γ Lκ 0 (X f − X s ) δ (ξ ) = 0 ⎠ ; X n(
where X s = X s / L
f)
= X n( ) / L f
; Xf = X f / L
[6.26]
; ξ = x/L.
For a tube terminated at the outlet by the dimensionless impedance α out , the displacement field of the fluid is written as the modal series: ∞
X f = ∑ X n( ) cos ( nπ + α out ) ξ f
[6.27]
n =0
Modal projection of [6.26] gives:
(κ μn
2 0
∞ ⎛ f ⎞ − κ 2 ) X s + γ Lκ 02 ⎜ X s − ∑ X n( ) ⎟ = 0 n =0 ⎝ ⎠
(( nπ + α
out
)
2
2
)
(f)
− κ Xn
⎛ ∞ ⎞ f + γ Lκ ⎜ ∑ X n( ) − X s ⎟ = 0 n = 0,1, 2,… ⎝ n =0 ⎠
[6.28]
2 0
where the modal coefficient μn is defined as: 1
μn =
⌠ μ f ⎮⎮ ⎮ ⌡0
( cos ( nπ + α ) ξ ) out
2
dξ
As a first example, the case α out = 0 is assumed, which implies:
[6.29]
Vibroacoustic coupling
⎧ μ f if n = 0 μn = ⎨ ⎩ μ f / 2 if n > 0
473
[6.30]
In the modal coordinate system, the displacement field of the fluid-structure system is defined as:
[Xva ]T
= ⎡⎣X s
X1(
f)
... X N( f ) ⎤⎦ f
[6.31]
N f is the order of truncation of the modal basis used to describe the fluid, and the
subscript
( va )
is used to identify the vibroacoustic nature of the quantity considered,
here the displacement field. With the aid of [6.30] and [6.31], the modal problem [6.28] takes on the canonical form which holds for any discrete and conservative mechanical system, written here in the reduced form as: ⎡⎣[Kva ] − κ 2 [Mva ]⎤⎦ [X va ] = [0]
[6.32]
The vibroacoustic stiffness and mass matrices [Kva ] and [M va ] are both symmetric.
[Mva ]
is positive definite and [Kva ] is positive. Furthermore, it is found convenient
to expand [Kva ] as the sum of the three following distinct matrices:
[Kva ] = [Ks ] + ⎡⎣K f ⎤⎦ + ⎡⎣K fs ⎤⎦
[6.33]
All these matrices are square and of the same size N × N , where N = 1 + N f . [K s ] is built with the stiffness matrix of the structure, including the elastic supports. In the present example, the only nonzero term is K s (1,1) = κ 02 . The related functional
[X va ]T [Ks ][X va ]
is that of the elastic energy of the structure and its supports. In the
same way ⎡⎣K f ⎤⎦ is built with the stiffness matrix of the fluid, including the elastic terminal impedance, which is α out = 0 in the present example. Thus ⎡⎣K f ⎤⎦ is diagonal and its coefficients are: K f (1,1) = 0 K f ( n + 1, n + 1) =
μ f ( n − 1) π 2 2
2
; n = 1, 2,… , N f
[6.34]
The related functional [X va ] ⎡⎣K f ⎤⎦ [X va ] is that of the elastic energy of the fluid. T
Finally, ⎡⎣K fs ⎤⎦ is the stiffness matrix which describes the fluid-structure interaction, modelled here as an elastic lumped element (cf. Chapter 5, subsection 5.3.2.4). All the terms except those of the first row and the first column are identical: K fs ( n1 , n2 ) = γ Lκ 02
n1 , n2 = 2,...1 + N f
[6.35]
474
Fluid-structure interaction
The terms of the first row and those of the first column are: K fs (1,1) = γ Lκ 02
; K fs (1, n ) = K fs ( n,1) = −γ Lκ 02
n = 2,...1 + N f
[6.36]
The related functional [X va ] ⎡⎣K fs ⎤⎦ [X va ] is that of the elastic energy of the spring which connects the structure and the fluid to each other. T
The mass matrix [M va ] is written as the sum of two distinct matrices:
[Mva ] = ⎡⎣[Ms ] + ⎡⎣M f ⎤⎦ ⎤⎦
[6.37]
The physical mass M s of the structure is used as a scaling factor to define the dimensionless mass matrix of the structure, denoted [M s ] and that of the fluid, denoted ⎡⎣M f ⎤⎦ . In the present example the only nonzero term of
[M s ]
is
M s (1,1) = 1 and ⎡⎣M f ⎤⎦ is diagonal: M f (1,1) = 0 ; M f ( n, n ) = μ n −1
n = 2,...1 + N f
[6.38]
The functionals [X va ] [M s ][X va ] and [X va ] ⎡⎣M f ⎤⎦ [X va ] are those of the kinetic energy of the structure and the fluid, respectively. Figure 6.4 displays the shapes of the first four vibroacoustic modes, as obtained by using the substructuring method (full line) and the exact method described in the last subsection (dotted line). At the scale of the plots, it would be difficult to detect any difference between the full and the dotted lines. However, small discrepancies occur at the fluid-structure interface, which are found to increase with the rank of the mode. Such errors can be made as small as desired by increasing N. On the other hand, the relative error on the reduced natural frequencies κ n is found to be less than one percent, provided the stiffness coefficient of the connecting element is selected according to the procedure described in [AXI 05], Chapter 4: T
1 1 2 = − γL 3 π2
N
1
∑n n =3
2
T
[6.39]
It is recalled that, in the context of longitudinal motions of beams, 1/ γ L stands for the flexibility coefficient of the neglected modes of the free-free modal basis. In the present context, it stands for the flexibility coefficient of the neglected acoustical modes of the tube open at both ends. Note that the value 1/3 refers to the static flexibility of the fluid column. The modal values of the functionals mentioned above are suitable to measure the relative importance of structural and fluid contribution to the vibroacoustic modes. va Starting from the mode shapes ⎡⎣X n( ) ⎤⎦ as expressed in the generalized coordinate system of the structural and acoustical modes and normalized with respect to the mass matrix by the usual condition:
Vibroacoustic coupling
μ n(
va )
T
va va = ⎣⎡X n( ) ⎦⎤ [Mva ] ⎣⎡X n( ) ⎦⎤ = 1
∀n
The relative contribution of the structure to μn(
475
[6.40] va )
is measured by:
T
s va va μn( ) = ⎡⎣X n( ) ⎤⎦ [M s ] ⎡⎣X n( ) ⎤⎦
[6.41]
That of the fluid is measured by: T
f va va μn( ) = ⎡⎣X n( ) ⎤⎦ ⎡⎣M f ⎤⎦ ⎡⎣X n( ) ⎤⎦
[6.42]
As shown in Figure 6.5, the relative part of the structural motion becomes rapidly negligible as the modal rank n is increased. Such a result explains why the modal displacement is found to be essentially zero at the fluid-structure interface for modes higher than n = 3, as it can be noticed in Figure 6.4.
Figure 6.4. The first four vibroacoustic modes of the mass-spring connected to a tube open at the outlet κ 0 = 1 and μ f = 1 . Displacement mode shapes: modal approximation N f = 20 (full line) and exact result (dotted line)
476
Fluid-structure interaction
Figure 6.5. Modal mass coefficients using a semilogarithmic scale (left plot) and a linear scale (right plot)
In the same way, Figure 6.6 shows the following elastic functionals: κ n(
va )
(f)
κn
T
T
va va = ⎡⎣X n( ) ⎤⎦ [Kva ] ⎡⎣X n( ) ⎤⎦ = κ n2 T
va va = ⎡⎣X n( ) ⎤⎦ ⎡⎣K f ⎤⎦ ⎡⎣X n( ) ⎤⎦
s va va ; κ n( ) = ⎡⎣X n( ) ⎤⎦ [K s ] ⎡⎣X n( ) ⎤⎦
( fs )
; κn
T
va va = ⎡⎣X n( ) ⎤⎦ ⎡⎣K fs ⎤⎦ ⎡⎣X n( ) ⎤⎦
[6.43]
Figure 6.6. Modal stiffness coefficients using a semilogarithmic scale (left plot) and a linear scale (right plot)
Again, and for the same reason, the contribution of the structure to the total elastic energy is found to decrease rapidly with n. On the other hand, the reader may va be puzzled by the fact that up to n = 4, the resultant functional κ n( ) is found to be less than the structural component κ n( ) . Explanation is that the fluid-structure s
interaction functional κ n( ) is not necessarily positive. As indicated in the right-hand side plots of Figure 6.6, in the present case it is negative up to mode n = 6, which is the reason why the corresponding line (dash-dotted line marked with stars) in the semilogarithmic plot, starts at n = 7, see also the right-hand linear plot. Figures 6.7 to 6.9 refer to the case of a closed tube ( α out = π / 2 ). The stiffness coefficient of the fs
Vibroacoustic coupling
477
connecting element which is appropriate for the tube open at one end and closed at the other is found to be (cf. [AXI 05], Chapter 4, page 235): 1 8 = 1− 2 γL π
N f +1
1
∑ ( 2n − 1) n =1
2
[6.44]
In Figure 6.7, it can be verified that the displacement of the piston, marked as an asterisk in the plots is essentially the same as that of the fluid at the fluid-structure interface. In this case also it is found that the vibroacoustic modes can be practically identified to the purely acoustical modes of the tube closed at both ends, as soon as n is higher than 2. As a final remark, it has been checked that if N = 30 modes are retained in the basis, the natural frequencies can be computed with an accuracy of about 0.1%.
Figure 6.7. The first four vibroacoustic modes of the mass-spring connected to a tube closed at the outlet, κ 0 = 1 and μ f = 1 . Displacement mode shapes: modal approximation (full line) and exact result (dotted line). The asterisk corresponds to the computed displacement of the piston
6.2.1.3 Modal expansion method: pressure as the fluid variable The modal expansion method can also be worked out starting from the mixed formulation [6.1], written for the piston fluid problem as:
478
(( K
Fluid-structure interaction
s
)
− ω 2 M s ) X s + pS f δ ( x ) = 0 2
[6.45]
d2p ⎛ ω ⎞ + ⎜ ⎟ p − ω 2 ρ f X sδ ( x ) = 0 dx 2 ⎝ ce ⎠
The lack of symmetry of the coupling terms can be immediately noticed, which indicates that the modal problem differs from those already solved up to now, from the mathematical standpoint at least. The pressure field is written as the modal series: ∞
p = ∑ pnϕ n( n =1
ϕ n(
p)
( x)
p)
( x)
[6.46]
are the acoustical mode shapes of the tube, expressed in terms of pressure.
A short inspection of [6.45] suffices to show that pressure cannot be identically zero at the fluid-structure interface; otherwise the fluid-structure system would not be coupled. Therefore, it is suitable to select a volume velocity node at the tube inlet ( x = 0 ). Hence the mode shapes are of the type (cf. Chapter 4, formula [4.74]): ϕn(
p)
⎛ ( ( 2n − 1) π − 2α out ) ξ ⎞ ⎟ ⎟ 2 ⎝ ⎠
(ξ ) = cos ⎜⎜
n = 1, 2,...N f
[6.47]
where ξ = x / L . As in the last subsection, before projecting the equations [6.45] onto the modal basis of each subsystem, they are transformed using the scaling factors [6.14] and the Young’s modulus of the fluid to scale the pressure, as well as the tube length to scale the displacement of the piston. The system projected onto the modal basis which comprises the single mode of the structure (mass-spring system) and N f acoustical modes of the fluid, is finally written in the following dimensionless form: Nf
(κ 02 − κ 2 ) X s + μ f ∑ϖ n = 0 n =1
2 ⎛ ⎛ ( 2n − 1) π ⎞ ⎞ ⎜⎜ − α out ⎟ − κ 2 ⎟ϖ n + 2κ 2X s = 0 ⎜⎝ ⎟ 2 ⎠ ⎝ ⎠
[6.48] n = 1,… N f
where the sign of the acoustic equation has been changed with respect to [6.45] and the following set of dimensionless parameters and variables are used: X s = X s / L ; ϖ n = pn / ρ f ce2 κ = ω L / ce
; μf = M f / Ms
; κ 0 = ω0 L / ce
The variables are used to define the so-called mixed displacement field as:
[6.49]
Vibroacoustic coupling
[Xva′ ]T
= ⎣⎡X s ϖ 1 … ϖ N f ⎦⎤
479
[6.50]
Once more, the modal problem [6.48] takes on a matrix form similar to [6.32], and of the same size (1 + N f ) × (1 + N f ) written as: ⎡⎣[Kva′ ] − κ 2 [Mva′ ]⎤⎦ [X va′ ] = [0]
[6.51]
where the mixed “stiffness matrix” [Kva′ ] is substituted for the stiffness matrix
[Kva ]
of the symmetric formulation in terms of structure and fluid displacement
fields, and the mixed “mass matrix”
[Mva ] . Matrix [Kva′ ]
[Mva′ ]
is substituted for the mass matrix
can be expanded as the sum of the three following matrices:
[Kva′ ] = ⎡⎣[Ks ] + ⎡⎣K f′ ⎤⎦ + ⎡⎣K fs′ ⎤⎦ ⎤⎦
[6.52]
[Ks ] is the stiffness matrix of the structure. Here, the only nonzero term is K s (1,1) = κ 02 . Matrix ⎡⎣K f′ ⎤⎦ is diagonal, whose components are the eigenvalues of the acoustical modal problem. The generic term is: ⎛ ( ( 2n − 1) π − α out ) ⎞ K f′ ( n + 1, n + 1) = ⎜ ⎟ ; n = 1, 2,… N f ⎜ ⎟ 2 ⎝ ⎠ 2
[6.53]
⎡⎣K fs′ ⎤⎦ describes the fluid-structure coupling terms entering into the structure equation. It is singular and non symmetric. In the present example, only the terms of the first row are non zero: K fs′ (1, n ) = μ f
n = 2,3,… , N f + 1
[6.54]
Matrix [Mva′ ] can also be expanded as the sum of three distinct matrices:
[Mva′ ] = ⎡⎣[Ms ] + ⎡⎣M f′ ⎤⎦ + ⎡⎣M fs′ ⎤⎦ ⎤⎦ [M s ] is the M s (1,1) = 1 .
[6.55]
mass matrix of the structure. Here, the only nonzero coefficient is ⎡⎣M f′ ⎤⎦ is a diagonal matrix, whose coefficients are:
M f′ ( n + 1, n + 1) = 1 ; n = 1, 2,… , N f
[6.56]
Finally, the matrix ⎡⎣M fs′ ⎤⎦ describes the fluid-structure coupling terms entering into the fluid equation, which is also singular and non symmetric, like ⎡⎣K fS′ ⎤⎦ . In the present example, only the terms of the first column are nonzero: M fs′ ( n,1) = −2 n = 2,3,… , N f + 1
[6.57]
480
Fluid-structure interaction
As ϖ n stands for a pressure, the matrices ⎡⎣ K ′f ⎤⎦ and ⎡⎣K fs′ ⎤⎦ have not the physical meaning of stiffness matrices while ⎡⎣M f′ ⎤⎦ and ⎡⎣M fs′ ⎤⎦ have not the meaning of mass matrices. The dimensionless functionals related to these matrices are: ∞
∞
′ = [X va′ ] ⎡⎣K fs′ ⎤⎦ [X va′ ] = μ f X s ∑ϖ n EKf′ = [X va′ ] ⎡⎣K f′ ⎤⎦ [X va′ ] = ∑ κ n2ϖ n2 ; EKfs T
T
n =1
n =1
∞
∞
[6.58]
′ = [X va′ ] ⎡⎣M fs′ ⎤⎦ [X va′ ] = −2X s ∑ϖ n EMf′ = [X va′ ] ⎡⎣M f′ ⎤⎦ [X va′ ] = ∑ κ ϖ ; EMfs T
n =1
2 n
2 n
T
n =1
Figure 6.8. First four vibroacoustic modes of the mass-spring connected to a tube open at the outlet, κ 0 = 1 and μ f = 1 .Pressure mode shapes: modal approximation (full line) and exact result (dotted line)
Generally, solving the non symmetrical system [6.45] requires the use of numerical methods such as those already described in [AXI 04], Appendices 5 and 8. Due to the lack of symmetry of the matrices, the Choleski decomposition scheme is replaced by the so-called LU algorithm. The results are illustrated in Figure 6.8 which displays the pressure mode shapes of the first four vibroacoustic modes of the piston-fluid system as obtained by solving numerically the modal problem [6.51] using twenty acoustical modes (full line) and by using the exact method described in the last subsection (dotted line). The results are found to be as accurate as those
Vibroacoustic coupling
481
produced by using the symmetrical formulation in terms of fluid displacement. The pressure mode shape of the first vibroacoustic mode is nearly linear as expected for a low frequency mode (κ < 1) . This remark leads us to emphasize, if necessary, that modelling vibroacoustic coupling includes automatically modelling inertial coupling as the asymptotic case κ << 1 corresponding physically to the low frequency, or the long wave range. Finally, it is interesting to revisit the closed tube problem already discussed in subsections 6.2.1.1 and 6.2.1.2, using now the pressure modal expansion, which brings out as a particular solution a coupled eigenmode at zero frequency. Assuming α out = π / 2 + nπ , the mode n = 0 arises as a non trivial solution of the acoustic modal problem, which corresponds to a uniform static pressure. When this particular mode is included in the expansion basis, as appropriate, the dimensionless equations [6.48] lead to the modal matrix equation: ⎡κ 02 − κ 2 ⎢ 2 ⎢ κ ⎢ 2κ 2 ⎢ 2 ⎢ 2κ ⎢ ⎢ 2 ⎣⎢ 2κ NOTE –
μf −κ 2 0 0 0
μf
μf
0 π −κ 2 0
0 0 2 4π − κ 2
0
0
2
⎤ ⎧ XS μf ⎥⎪ϖ 0 ⎥⎪ 0 ⎥ ⎪⎪ ϖ 1 0 ⎥⎨ 0 ⎥ ⎪ ϖ2 ⎥⎪ ⎥⎪ N f 2π 2 − κ 2 ⎥⎦ ⎩⎪ϖ N f
⎫ ⎧0 ⎫ ⎪ ⎪ ⎪ ⎪ ⎪0 ⎪ ⎪⎪ ⎪⎪0 ⎪⎪ ⎬=⎨ ⎬ ⎪ ⎪0 ⎪ ⎪ ⎪0 ⎪ ⎪ ⎪ ⎪ ⎭⎪ ⎪⎩0 ⎪⎭
[6.59]
Vibroacoustic mode at zero frequency
The second row of this system, which is related to the uniform pressure mode, is of pure inertial nature. This leads to the occurrence of a peculiar vibroacoustic mode at zero frequency, which does not appear as a solution of equation [6.13], nor of equation [6.32]. It is however consistent with the mixed modal formulation using displacement and pressure. The corresponding mode shape is such that the uniform internal pressure is balanced by the deformation of the piston elastic support, as can be verified by solving equation [6.59]. Assuming κ = 0 and ϖ 1 = ϖ 2 = ... = ϖ N f = 0 , it follows that: 2
M p ⎛ω L⎞ X κ 02X s + μ f ϖ 0 = 0 ⇒ ⎜ 0 ⎟ s + f 0 = 0 ⇒ K s X s + S f p0 = 0 c L Ms Ef ⎝ e ⎠
[6.60]
Apart from this specificity, the modal results stemming from [6.59] are identical to those presented in the previous subsections. 6.2.1.4 Pressure and displacement potential as two fluid variables Equations [6.9] are particularized to the piston-fluid system as:
482
(( K
Fluid-structure interaction
s
)
− ω2M s ) X s − ω2S f Π δ ( x) = 0
⎛ S d Π Sf p ⎞ ω2 ⎜ − f + − ω 2 S f X sδ ( x ) = 0 ⎜ ρ dx 2 ρ c 2 ⎟⎟ f f e ⎠ ⎝
[6.61]
2
By complementing the system [6.61] with the relation of definition [6.6], the fluidstructure modal problem can be formulated in a self-adjoint (symmetric) matrix form as: ⎡( K s − ω 2 M s ) δ ( x ) ⎢ ⎢ −ω 2 S f δ ( x ) ⎢ 0 ⎢ ⎣
−ω 2 S f δ ( x )
( −ω S 2
f
⎤ ⎥ ⎡ X s ⎤ ⎡0⎤ +ω 2 S f / ρ f ce2 ⎥ ⎢ Π ⎥ = ⎢0⎥ ⎥⎢ ⎥ ⎢ ⎥ + S f / ρ f ce2 ⎥ ⎢⎣ p ⎥⎦ ⎢⎣0⎥⎦ ⎦ 0
/ ρ f ) d 2 / dx 2
+ω 2 S f / ρ f ce2
[6.62]
which is of the same canonical form as the solid part of the problem, except that the physical meaning of the variable field and of the operators differs: ⎡⎣[ K va′′ ] − ω 2 [ M va′′ ]⎤⎦ [ X va′′ ] = [0]
[ K va′′ ]
[6.63]
is the singular and diagonal matrix:
⎡ K sδ ( x ) 0 ⎤ 0 ⎢ ⎥ 0 0 [ K va′′ ] = ⎢ 0 ⎥ ⎢ 0 0 S f / ρ f ce2 ⎥⎦ ⎣
[ M va′′ ]
[6.64]
is the symmetric matrix:
⎡ M sδ ( x ) ⎢ [ M va′′ ] = ⎢ S f δ ( x ) ⎢ ⎣ 0
(S
S fδ ( x)
f
/ ρ f ) d 2 / dx 2 2 e
−S f / ρ f c
⎤ ⎥ − S f / ρ f ce2 ⎥ ⎥ 0 ⎦ 0
[6.65]
Another important point worth emphasizing again is that since p and Π are not independent fields, no new degrees of freedom are introduced into the physical system. Nevertheless, in agreement with the formalism of the Lagrange multipliers, from the mathematical standpoint Π is treated as an additional independent field which is constrained by the rheonomic condition [6.6]. As a consequence, the number of degrees of freedom of the constrained system is modified, as further described below. The energy functional related to [ K va′′ ] is:
Vibroacoustic coupling
EKva =
L ⌠ ⎮ ⎮ ⎮ ⎮ ⎮ ⎮ ⎮ ⌡0
[Xs
⎡ K sδ ( x ) 0 0 ⎤ ⎡Xs ⎤ S Lp 2 ⎢ ⎥ p] ⎢ 0 0 0 ⎥ ⎢ Π ⎥ dx = K s X s2 + f ⎢ ⎥ Ef ⎢ 0 ⎥ ⎣⎢ p ⎦⎥ 0 / S E f f ⎣ ⎦
Π
483
[6.66]
p 2 is the mean square value of pressure, defined as: p2 =
L
1⌠ 2 ⎮ p dx L ⎮⌡0
[6.67]
Hence, as could be expected, the functional [6.66] is found to be the total elastic energy of the fluid-structure system. It is independent of Π and nil for any steady fluid motion about the fixed structure, since in such cases both the displacement field of the structure and the fluctuating pressure field in the fluid are identically zero. By definition, the energy functional related to [ M va′′ ] is:
EM va′′ =
L ⌠ ⎮ ⎮ ⎮ ⎮ ⎮ ⎮ ⎮ ⎮ ⌡0
[Xs
⎡ M sδ ( x ) ⎢ p] ⎢ S f δ ( x ) ⎢ ⎣ 0
Π
(S
S fδ (x)
f
/ ρ f ) d / dx 2
⎤⎡X ⎤ ⎥ s − S f / E f ⎥ ⎢ Π ⎥ dx ⎢ ⎥ ⎥ 0 ⎦ ⎢⎣ p ⎥⎦ 0
2
−S f / E f
[6.68]
Assuming that Π complies with self-adjoint boundary conditions defined as: aΠ + b
dΠ dx
=0
[6.69]
0, L
EM va′′ can be expressed as the symmetric form: EM va′′ = M s X + 2 S f X s Π ( 0 ) 2 s
L ⌠ ⎮ − ⎮⎮ ⎮ ⎮ ⌡0
⌠
L
⎮ S Π p S f ⎛ d Π ⎞2 f ⎮ dx ⎜ ⎟ dx − 2⎮ ρ f ⎝ dx ⎠ Ef ⎮
[6.70]
⌡0
2
As could be expected, the quantity ω EM va′′ has the dimension of a kinetic energy. It is also of special interest to derive a dimensionless formulation of the coupled problem which allows ones to adjust the appropriate order of magnitude of the coefficients of the matrices in relation to the physical importance of the fluid and the structure terms. With this object in mind, we start by defining the dimensionless field variables: ξ=
x L
; Xs =
Xs L
; φ=
Π Φ = ρ f L2 L2
; ϖ=
φ is the dimensionless displacement potential.
p Ef
; μf =
Mf Ms
[6.71]
484
Fluid-structure interaction
With the aid of the reduced quantities [6.14] and [6.71], the matrix equation [6.62] is rewritten as: ⎡ (κ 02 − κ 2 ) δ (ξ ) − μ f κ 2δ (ξ ) ⎢ 2 −μ f κ 2d 2 / dξ 2 ⎢ − μ f κ δ (ξ ) ⎢ +μ f κ 2 0 ⎢⎣
⎤ ⎡X ⎤ ⎡0⎤ ⎥ s + μ f κ ⎥ ⎢ φ ⎥ = ⎢ 0⎥ ⎢ ⎥ ⎢ ⎥ + μ f ⎥ ⎢⎣ ϖ ⎥⎦ ⎢⎣0⎥⎦ ⎥⎦ 0
2
[6.72]
The system can be discretized using either the modal expansion method, or the finite element method. According to the first method, the φ and ϖ fields are expanded as: ∞
φ = ∑ φnϕ n( n =1
p)
∞
; ϖ = ∑ϖ nϕ n(
(ξ )
n =1
p)
(ξ )
[6.73]
where the pressure mode shapes [6.47] are used. Modal projection leads to the quadratic form:
(κ
2 0
∞ ∞ ⎛ ⎞ − κ 2 ) X s2 + bnϖ n2 − κ 2 μ f ⎜ 2X s ∑ φm − ∑ amφm2 + 2bmφmϖ m ⎟ m =1 m =1 ⎝ ⎠
[6.74]
The modal coefficients an and bn are:
an =
bn =
⎧ ( 2n − 1)2 π 2 (open outlet) ⎪− dϕ n ( ξ ) ⎞ ⎪ 8 ⎟ dξ = ⎨ 2 ⎟ 2 dξ ⎪ ( n − 1) π ⎠ − (closed outlet) ⎪⎩ 2 1 ⎧ (open outlet) ⎪ 2 ⎪ p 2 ϕ n( ) (ξ ) d ξ = ⎨ ⎪b1 = 1, bn >1 = 1 (closed outlet) ⎪⎩ 2
1 ⌠ ⎮ ⎮ ⎛ −⎮ ⎜ ⎮ ⎜ ⎮ ⎝ ⌡0 1 ⌠ ⎮ ⎮ ⎮ ⌡0
(
2
( p)
[6.75]
)
To describe the major features of the modal problem to be solved, it suffices to consider a reasonably low sized algebraic modal equation, for instance that resulting from a discretization comprising four degrees of freedom, which leads to the 7 × 7 matrix equation: ⎡ κ 02 − κ 2 ⎢ 2 ⎢ −μ f κ ⎢ 0 ⎢ 2 ⎢ −μ f κ ⎢ 0 ⎢ 2 ⎢ −μ f κ ⎢ 0 ⎣
−μ f κ 2 2
− μ f κ a1 + μ f κ 2b1
0 + μ f κ 2b1 + μ f b1
−μ f κ 2
0
−μ f κ 2
0 0 + μ f κ 2 b2 + μ f b2
0 0
0 0
0
0
0
0
0 0 − μ f κ 2 a2 + μ f κ 2b2
0 0
0 0
0 0
0 0 − μ f κ 2 a3 + μ f κ 2 b3
⎤ ⎡X s ⎤ ⎡ 0⎤ ⎥⎢ ⎥ ⎢ ⎥ 0 ⎥ ⎢ φ1 ⎥ ⎢ 0⎥ ⎥ ⎢ϖ 1 ⎥ ⎢ 0⎥ 0 ⎥⎢ ⎥ ⎢ ⎥ 0 ⎥ ⎢ φ2 ⎥ = ⎢ 0⎥ ⎥ ⎢ϖ ⎥ ⎢ 0⎥ 0 2 ⎥ ⎢ ⎥ ⎢ ⎥ + μ f κ 2b3 ⎥ ⎢ φ3 ⎥ ⎢ 0⎥ + μ f b3 ⎥⎦ ⎢⎣ϖ 3 ⎥⎦ ⎢⎣ 0⎥⎦ 0
[6.76]
Vibroacoustic coupling
485
The “mass” matrix is: ⎡1/ μ f ⎢ 1 ⎢ ⎢ 0 [Mva′′ ] = μ f ⎢⎢ 1 ⎢ 0 ⎢ ⎢ 1 ⎢ 0 ⎣
1
0
a1 −b1 0 0
−b1 0 0 0
0 0
0 0
1 0
0 0
1 0
0
0
a2 −b2 0 0
−b2 0
0 0 0
0 0
a3 −b3
0 ⎤ 0 ⎥ ⎥ 0 ⎥ ⎥ 0 ⎥ 0 ⎥ ⎥ −b3 ⎥ 0 ⎥⎦
[6.77]
As expected, the “stiffness” matrix is singular: ⎡κ 02 / μ f ⎢ ⎢ 0 ⎢ 0 ⎢ K va′′ = μ f ⎢ 0 ⎢ 0 ⎢ ⎢ 0 ⎢⎢ 0 ⎣
0 0 0 0 0 0 0
0 0 b1 0 0 0 0
0 0 0 0 0 0 0
0 0 0 0 b2 0 0
0 0 0 0 0 0 0
0 0 0 0 0 0 b3
⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎥ ⎦
[6.78]
It has as many zero eigenvalues as there are acoustical modes included in the model. The structural displacement and the pressure components of the corresponding eigenvectors are necessarily zero, and the components of the displacement potential Φ can be selected arbitrarily. As pointed out above, such mathematical solutions arise because Π is treated as an additional independent field, feeding thus the system with its own degrees of freedom; and because the elastic energy does not depend on Π . As there is no kinetic energy associated with such motions, they arise as non trivial solutions of the modal problem [6.76]. More generally, denoting N f the number of acoustical modes retained in the modal basis and N s that of the structural modes, the modal fluid-structure problem comprises N = N s + 2 N f solutions, N s + N f being physically meaningful, corresponding to real motions of the coupled system. The remaining N f eigenmodes are at zero frequency, zero pressure and solid displacement. Hence, they are of no interest from the physical standpoint. Results arising from the modal expansion method based on the ( X s , Π , p ) formulation are further discussed in relation to those obtained by using the two displacements and the mixed ( X s , p ) methods, in subsection 6.4.1.1, using an illustrative example.
486
6.2.2
Fluid-structure interaction
Analytical solutions of forced problems
It is of practical interest to assess qualitatively the importance of modelling vibroacoustic coupling before proceeding to a quantitative analysis of the fluidstructure system one is interested in. The following examples have been especially selected to point out several major features of the response in relation with the order of magnitude of a few dimensionless parameters. The importance of the mass ratio μ f already introduced in the modal problems, that of the stiffness ratio γ f = K f / K s , where K f is the characteristic stiffness coefficient of the fluid and K s that of the structure, and finally that of the dimensionless frequency or wave
number κ = ω L / ce are emphasized in relation to the characteristics of excitation and response of the coupled system. 6.2.2.1 Piston coupled to a fluid column and forced harmonically As a first exercise, the steady vibration of the piston-fluid system depicted in Figure 6.9 is analysed. The piston is modelled as a slightly damped mass-spring system driven by a harmonic force F0 eiω t . The straight tube of constant crosssectional area and filled with a compressible fluid is terminated at the outlet by the acoustic impedance α out . The problem is solved first by adapting the local equation [6.12] to the present problem. It reads as: ⎛ ⎛ωL ⎞⎞ ω ce 2 + αout ⎟ ⎟⎟ X s (ω ) = F0 M f tan ⎜ ⎜⎜ K s + iωCs − ω M s − L ⎝ ce ⎠⎠ ⎝
[6.79]
Figure 6.9. Spring mass damper system coupled to a compressible fluid column in a tube
Solution is immediate. Defining the damping ratio of the structure as: ζ0 =
Cs Cs = 2 M s K s 2 M sω0
The complex amplitude of the tube displacement is written as:
[6.80]
Vibroacoustic coupling
Fo Ms
X (κ ) =
⎛L⎞ ⎜ ⎟ ⎝ ce ⎠
2
⎛ ⎞ 1 ⎜ 2 ⎟ 2 ⎜ ⎡κ 0 − κ + 2iζ 0κ 0κ − μ f κ tan (κ + αout ) ⎤ ⎟ ⎣ ⎦ ⎝ ⎠
487
[6.81]
The result [6.81] can also be expressed as: X (κ ) X0
=
1 κ κ μfκ − 1 − 2 + 2iζ 0 tan (κ + α out ) κ 0 κ 02 κ0 2
[6.82]
X 0 = F0 / K s is the static displacement of the piston induced by the static force F0 . As long as damping is sufficiently small, the magnitude of the frequency response [6.82] is sharply peaked at the vibroacoustic resonances. Recalling that the latter are the roots of equation [6.13], the magnitude of the resonant responses is given by: X (κ n ) κ0 ω0 = = X0 2ζ 0κ n 2ζ 0ωn
[6.83]
It is of interest to investigate the spectral response of the piston as produced by three distinct models, namely the consistent vibroacoustic model described just above and the simplistic models in which the liquid is either discarded, or accounted for by adding its physical mass M f to that of the structure. The results are presented as plots of the spectral power density SXX ( f ) of the displacement of the piston, expressed in mm 2 /Hz and as the r.m.s. value σ XX expressed in mm. By definition (see for instance [AXI 04], Chapter 8): ∞
σ XX =
⌠ ⎮ ⎮ ⌡0
SXX ( f ) df
[6.84]
In this respect, it is useful to recall that the response of a slightly damped harmonic oscillator is given to a high degree of accuracy by the following formula (see [AXI 04], Chapter 9): σ XX = X 0
π 2ς 0
[6.85]
The results presented here refer to the following numerical parameters: 1. Fluid column: Fluid density: ρ f = 1 000 kgm -3 ; effective speed of sound: ce = 1 000 ms-1 ; tube radius: R = 12.5 cm; wall thickness: e = 4 mm; tube length: L = 24 m; scaling factor of the stiffness of the fluid column: K f = ρ f ce2 S f / L 2106 N/m . Mass of the fluid column: M f = π R 2 Lρ f = 1178 kg .
488
Fluid-structure interaction
2. Structure: Mass-spring system:
M s = 588 kg
( μ f = 2 ), damping ratio in vacuum
ζ 0 = 0.04 , stiffness coefficient of the spring K s , varied from 0.1K f to 10 K f . The
mass M s of the piston is equal to that of the tube in vacuum. On the other hand, the damping ratio of the tube with the mass of fluid added to it is ζ i = 2.3110−2 . 3. Excitation The excitation signal is F0 eiω t , with F0 = 1 kN . It can be interpreted either as a harmonic force as suggested above, or as the Fourier transform of an impulsive load. The spectra shown in Figure 6.10 are obtained by supposing that the tube is open at the outlet. The dashed line refers to the model in vacuum and the corresponding r.m.s value is denoted σ v . The dotted line refers to the incompressible model and the corresponding r.m.s value is denoted σ i . Finally, the full line refers to the compressible model and the corresponding r.m.s value is denoted σ c . The upper left plot corresponds to a quasi incompressible case (stiffness ratio γ f = K f / K s equal to 10). Accordingly, the first vibroacoustic peak merges practically with that of the incompressible fluid model. The r.m.s displacements, as calculated by integrating the spectrum, are also the same in both models and larger than the value of the model in vacuum by the quantity ς 0 / ς i , in full agreement with formula [6.85]. As γ f is decreased, the frequency of the first vibroacoustic resonance lags behind that
of the incompressible fluid model, in agreement with the behaviour already discussed in subsection 6.2.1.1, and several other vibroacoustic resonances enter into the frequency range explored, which are however predominantly acoustic in nature. Such modes are easily identified by inspecting the resonance peaks of the plots, which are much sharper than the other, indicating that damping ratio of predominantly acoustic modes is much less than that of the structural or predominantly structural modes. The reason for such behaviour is that viscous damping force is assumed to be proportional to the velocity of the structure, whereas fluid is supposed lossless (conservative model). Another remarkable result is that even if the spectral response of the piston is highly modified with respect to those produced by the simplistic in vacuum or added mass models, the r.m.s displacement of the compressible model remains the same as that produced by the added mass model, even if the compressibility effect is very large. As a consequence, whatever the value of γ f may be, σ XX remains simply proportional to 1/ K s , and to 1/ ς , as predicted by formula [6.85].
Vibroacoustic coupling
489
Figure 6.10. Spectral response of the piston to a harmonic force swept in frequency, tube open at the outlet α out = 0 . Full line: vibroacoustic model, dotted line: incompressible fluid model, dashed line: no fluid
Let us turn now our attention to the spectra shown in Figure 6.11, which correspond to a tube closed at the outlet. As a consequence, if γ f is very large, the fluid is essentially rigid and the piston is practically blocked. Obviously, both the in vacuum and the added mass models are irrelevant to describe such a physical situation. In practice, as long as γ f is sufficiently large, the r.m.s displacement is substantially less than the values predicted by the simplistic models mentioned just above. However as γ f is decreased, because fluid stiffness becomes less and less important in comparison with that of the structure, σ XX is found to tend asymptotically toward the value predicted by the in vacuum model. As a conclusion of the exercise, it is found that the amplitude of the response to a harmonic force of the spring-mass system coupled to a fluid column depends critically on the impedance seen by the fluid at the outlet. If the impedance is low, the r.m.s amplitude of the displacement does depend directly neither on the mass nor on the stiffness ratios μ f and γ f . Indirect dependency through the damping ratio remains possible but cannot be investigated without an appropriate model of dissipation, which is out the scope of the present study. On the other hand, if the
490
Fluid-structure interaction
outlet impedance is high, the magnitude of the response highly depends on the stiffness ratio γ f , decreasing and tending to zero as γ f is increased.
Figure 6.11. Spectral response of the piston to a harmonic force swept in frequency, tube closed at the outlet α out = π / 2 . Full line vibroacoustic model, dotted line: incompressible fluid model, dashed line: no fluid
6.2.2.2 Response to a transient force exerted on the piston The problem is formulated as follows: K s X s + M s X s + p ( 0; t ) S f = F (
e)
(t )
∂ p 1 ∂ p − =0 ∂ x 2 ce2 ∂ t 2 2
∂p ∂x
2
= − ρ f X s
[6.86] ;
p( L; t ) = 0
x =0
The system is supposed at rest at times t ≤ 0 and F (
e)
(t )
stands for some transient
force acting on the piston at times t > 0 . The tube is supposed open at the right end. Laplace transformation of the system [6.86] leads to:
Vibroacoustic coupling
X s ( s ) =
e F ( ) ( s )
M s (ω + s 2 0
2
)
p (0; s ) S f
−
491
[6.87]
M s (ω02 + s 2 )
Note that here s means the Laplace variable and not a curvilinear abscissa. On the other hand, solving the wave equation in [6.86] is straightforward and, after a few elementary manipulations, the Laplace transform of the pressure field is found to be: p ( x; s ) =
− sρ f ce X s ( s ) 1 + eTa s
(e
sx / ce
− e s (Ta − x / ce )
)
[6.88]
Ta = 2 L / ce is the back and forth travel time of the acoustic waves within the tube. Substituting the pressure field [6.88] into the structure equation [6.87], the displacement of the piston can be related to the external excitation by the following relation: ⎛ X s ( s ) ⎜ M s (ω02 + s 2 ) + sρ f ce S f ⎝
⎛ 1 − e − sTa ⎜ − sTa ⎝1+ e
⎞ ⎞ ( e) ⎟⎟ = F (s) ⎠⎠
[6.89]
Relation [6.89] is quite remarkable for its physical content, as it shows that the fluid acts on the piston as a force proportional to the Laplace variable s, which marks a time derivative, and proportional to a series of exponential terms which characterize the periodic repetition of it, which is physically related to the back and forth travels of the sound waves. Restricting the analysis to the time interval 0 ≤ t ≤ Ta , delayed components of the response can be discarded and [6.89] takes on the canonical form of the response of a damped harmonic oscillator to an external transient force: X s ( s ) =
e F ( ) ( s )
M s (ω02 + 2 sω0ς va + s 2 )
[6.90]
The oscillator corresponds to the mass-spring system in vacuum provided with the viscous damping ratio: ς va =
ρ f ce S f 2 M s ω0
[6.91]
As astonishing as such a result may appear, at first sight at least, it is in fact quite natural. Physical explanation is in terms of mechanical energy delivered to the sound waves by the vibrating structure. Due to vibroacoustic coupling, part of the energy imparted to the piston by the external force is transferred to sound waves and some amount of mechanical energy is lost by the oscillator which is precisely equal to that radiated by the sound waves. As a consequence, the vibroacoustic coupling mechanism includes necessarily a dissipative force. As further established in a more systematic way in Chapter 7, it turns out that in the particular case of plane waves such a force fits precisely to the viscous damping model, that is, the dissipative force
492
Fluid-structure interaction
is proportional to the velocity of the piston. This however is only one part of the story, since if the pipe outlet is terminated by a conservative impedance, the waves are entirely reflected back and the radiated energy is returned to the oscillator. As a consequence, amplitude of the response is periodically restored, at the long period Ta of back and forth travels of the sound waves. As a consequence, the rate of exchange of energy between the structure and the fluid is expected to depend on the ratio of the characteristic times of motion in both media, which turns out to be here the period of the mass-spring system and the travel time of the wave through the fluid column. Actually, the vibroacoustic damping ratio [6.91] can be easily transformed to put this ratio in evidence. Substituting the relation K s = ω02 M s into [6.91], we obtain the alternative forms: ς va =
ρ f ce S f 2 M s ω0
=
ρ f S f Lce 2 M s Lω0
=
μf 2κ 0
=
μ f T0 2π Ta
[6.92]
It can be anticipated that the dynamical behaviour of the vibroacoustic system is highly dependent of the value of this ratio. As a first extreme case, T0 can be much smaller than Ta . Accordingly, the solid vibrates many times before receiving back the energy given to the waves. Consequently, the motion is expected to be a series of damped vibrations modulated in amplitude at the large time scale Ta , due to the periodical energy feedback by the waves. On the other hand the mass ratio is a measure of the transfer of energy to the fluid, as it can be interpreted as the ratio of the kinetic energy imparted to the fluid on that of the piston, as an order of magnitude at least. Therefore the higher is μ f the larger is ς va . Conversely, if T0 is much larger than Ta , it would be totally erroneous to conclude from [6.92] that the system is overdamped, and if so not oscillating anymore. This precisely because the whole amount of energy delivered to the fluid to set it in motion is given back to the structure at a time scale much shorter than T0 . Therefore, the oscillator behaves as a conservative system coupled to an incompressible fluid. Hence, if Ta T0 , the vibroacoustic coupling is expected to reduce practically to the inertial coupling mechanism, in full agreement with the conclusion derived from the usual criterion stated as κ 0 = ω0 L / ce << 1 . Inversion of the Laplace transform at times exceeding Ta is uneasy because of the presence of the delayed terms in the damping force and it is preferred to solve this type of problem by numerical integration, as illustrated on a few other examples in the next subsections.
Vibroacoustic coupling
493
6.2.2.3 Tube excited by a transient pressure source
Figure 6.12. Tube terminated by a closed end and excited by pressure source at the other
Figure 6.12 describes a system which is conceptually identical to the piston-fluid column though it differs physically. It consists of a uniform tube open at the inlet and closed at the outlet, containing some compressible fluid. Again, the system is e supposed at rest at times t ≤ 0 and P ( ) ( t ) δ ( x ) stands for some external pressure source concentrated at the inlet. The tube is assumed to be rigid. Its mass is M s and K s denotes the stiffness coefficient of the supports in the longitudinal direction. The problem can thus be formulated as follows: K X + M X = p ( L; t ) S s
s
s
2
s
f
2
∂ p 1 ∂ p − =0 ∂ x 2 ce2 ∂ t 2 p(0; t ) = P (
e)
(t )
[6.93] ;
∂p ∂x
= − ρ f X s x=L
The Laplace transform of the tube displacement is: p ( L; s ) S f X s ( s ) = M s (ω02 + s 2 )
[6.94]
Again solving the acoustic equation is straightforward and details are omitted. The Laplace transform of the pressure field at the closed end is found to be: p ( L; s) =
2 P (
(s
2
e)
( s ) ( s 2 + ω02 ) e− sT / 2 a
+ ω02 + 2ω0ς va s ) + ( s 2 + ω02 − 2ω0ς va s ) e − sTa
[6.95]
494
Fluid-structure interaction
Once more Ta = 2 L / ce is the back and forth travel time of the acoustic waves and ς va is the vibroacoustic damping ratio defined by relations [6.92]. Substituting [6.95] into [6.94] the displacement follows as: X s ( s ) =
Ms
(( s
2 P ( e ) ( s ) S f e − sTa / 2
2
+ ω02 + 2ω0ς va s ) + ( s 2 + ω02 − 2ω0ς va s ) e − sTa
)
[6.96]
It is noticed that the denominator of the Laplace transforms [6.95] and [6.96] are the sum of two distinct terms closely related to each other. The first term describes a damped oscillator whereas the second term describes the same oscillator with the damping term changed in sign. Furthermore the second term is multiplied by the exponential shift factor which transforms into the time domain as a delay precisely equal to Ta . During the time interval 0.5Ta < t < 1.5Ta , formulas [6.94] and [6.96] simplify into: e 2 P ( ) ( s ) ( s 2 + ω02 ) e− sTa / 2 p ( L; s ) = [6.97] s 2 + ω02 + 2ω0ς va s X s ( s ) =
2 P (
e)
( s ) S f e − sT / 2 a
[6.98]
M s ( s + ω + 2ω0ς va s ) 2
2 0
With the aid of equations [6.92], [6.96] and [6.98], the pressure at the closed end can be expressed as the sum of the two following distinct components: e p ( L; s ) = 2 P ( ) ( s ) e − sTa / 2 − ρ c sX ( s ) [6.99] f
e
s
The first component stands for the pressure wave triggered by the external source which travels through the fluid column as if the tube was at rest. The second component is proportional to the velocity of the tube. It stands for the vibroacoustic coupling field induced by the motion of the tube. To explain further the response of the coupled system it is necessary to assume first some analytical form for the excitation signal, sufficiently simple for easy calculation of the inverse Laplace transform of [6.97] and [6.98]. Let us consider for instance the case of an abrupt e change in static pressure, idealized as the step function P ( ) U ( t ) . If restricted to the time interval 0.5Ta < t < 1.5Ta , the Laplace transform of the tube displacement is: X s ( s ) =
2 P ( ) S f e − sTa / 2 e
M s s ( s 2 + ω02 + 2ω0ς va s )
[6.100]
The inverse Laplace transform can be determined analytically by using the method of residues, as explained in [AXI 04], or directly starting from the impulsive response, or Green function, of a damped oscillator. Assuming that the damping ratio ς va is smaller than the critical value ( ς va < 1 ), the inverse Laplace transform of [6.96] is obtained as the following time integral:
Vibroacoustic coupling
X s (θ ) =
2P( ) S f e
ωd M s
2P( )S f e
θ
⌠ ⎮ ⎮ ⌡0
e
− ω0ς va t
sin ωd t dt =
ωd2 M s
u / ω1
⌠ ⎮ ⎮ ⎮ ⎮ ⌡0
e
⎛ ζ u − ⎜ va ⎜ 1−ζ 2 va ⎝
⎞ ⎟ ⎟ ⎠
sin u du
495
[6.101]
θ = t − 0.5Ta is the retarded time which accounts for the delay needed by the excited pressure wave to reach the closed bottom of the tube, hence to excite the tube. On the other hand ωd is the so-called damped frequency defined as: ωd = ω0 1 − ς va2
[6.102]
Recalling the mathematical identity: ⌠ ⎮ ⎮ ⌡
e ax sin x dx =
e ax ( a sin ax − cos x ) 1 + a2
[6.103]
The time history of the tube displacement is finally found to be: X s (θ ) =
e 2 P ( ) S f ⎧⎪ −ω ς vaθ ⎨1 − e 0 Ks ⎪ ⎩
⎛ ⎞ ⎫⎪ 1 ⎜ sin ωd θ + cos ωd θ ⎟ ⎬ ⎜ 1−ς 2 ⎟⎪ va ⎝ ⎠⎭
[6.104]
Pressure at the closed outlet follows as: μ f −ω0ς vaθ ⎫ e ⎧ p (θ ) = 2 P ( ) ⎨1 − e sin ωd θ ⎬ ω d ⎩ ⎭
[6.105]
Of course validity of solutions [6.104] and [6.105] is restricted to the time interval 0 ≤ θ ≤ Ta before any wave reflection at the tube bottom. At later times, it is necessary to add the retarded terms whose contribution occur at the period Ta . The task is as difficult as in the first example and will not be attempted. On the other hand, it is also of interest to consider the range ς va >> 1 , which means either that the fluid mass is much larger than that of the structure, or that the acoustic waves are reflected many times back and forth during the time the structure makes a single cycle of oscillation, in which case it is expected that fluid compressibility can be neglected. If such is really the case, the problem can be formulated as follows: K X + M X = p ( L; t ) S s
s
s
s
f
2
∂ p =0 ∂ x2 p(0; t ) = P ( e ) ( t )
[6.106] ;
∂p ∂x
= − ρ f X s x=L
Solution is immediate. The pressure field is expressed as:
496
Fluid-structure interaction
p ( x; t ) = P (
e)
( t ) − ρ f Xs x
⇒
p( L, t ) = P (
e)
( t ) − ρ f Xs L
[6.107]
The forced oscillator equation is: e K s X s + ( M s + M f ) X s = P ( ) ( t ) S f
[6.108]
The response to a pressure source varying as a step function is: X s (t ) =
P( ) S f e
Ks
(1 − cos ω1t )
ω1 = K s / ( M s + M f
)
[6.109]
and once more M f = ρ f S f L is the mass of fluid contained
in the tube. Now the interesting point is to validate such a solution by recovering it as the asymptotical behaviour of the compressible solution when the travel time of the waves tends to zero. With that purpose in mind, it is necessary to start from solution [6.96], expressed in the domain of the Laplace variable, and not from the solutions [6.104] and [6.105], since the range of validity of the latter tends to zero as Ta tends to zero. Multiplying the numerator and denominator of [6.96] by exp ( sTa / 2 ) and expanding the exponential terms as a Taylor series limited to the
first order, we obtain: ⎛ 2 P(e) S f X s ( s ) = ⎜ ⎜ MS ⎝
e ⎞ ⎞⎛ P( ) S f 1 ⎟ = ⎟⎜ ⎟ ⎜ 2 ( s 2 + ω02 ) + 2 μ f s 2 ⎟ ( M s + M f ) ( s 2 + ω12 ) ⎠⎝ ⎠
[6.110]
Expression [6.110] identifies with the incompressible solution as it should. In Figure 6.13 two distinct time histories of the tube displacement are plotted. Numerical parameters of the computation are as follows: M s = 628 kg ; M f = 1570 kg ; L = 50 m ; ce = 500 m/s The plot on the left refers to the case f 0 20 Hz , so ς va is less than unity and the tube oscillates several times during the wave travel time Ta . Comparing the time history predicted by using the compressible model (full line) to that predicted by the incompressible model (dashed line), important differences are highlighted. In particular, the compressible model accounts for the propagation delay of the exciting wave, as expected. Then the frequency of the oscillations is larger than that predicted by the incompressible model and maximum amplitude is also significantly larger. Finally, the damping related to the energy given by the structure to the sound waves is very significant. The right-hand side plot is relative to a quasi incompressible case ς va 4.5 . The responses predicted by both models agree with each other in essence though some differences related to the propagation delay are still apparent.
Vibroacoustic coupling
497
Figure 6.13. Tube displacement plots, incompressible fluid in dashed line and compressible fluid in full line. In the last case, the solution is restricted to the time interval:
0.5Ta ≤ t ≤ 1.5Ta T0
6.2.3
Expansion methods to solve forced problems
Although the Laplace transform technique is an efficient technique to point out the important features of the vibroacoustic responses to various kinds of external excitations, difficulties often arise in expressing the final results in the time domain, due to the contribution of the reflected waves, as illustrated by the preceding examples. Therefore, it is generally preferred in practice, and even necessary, to solve the problem using a numerical method, based on the discretization of the partial derivative equations. In this subsection we shall be concerned with the modal expansion methods already introduced in subsection 6.2.1, in the context of modal problems. Furthermore, as we are mainly interested in describing the motion of the mass-spring system and not the propagation of the acoustic waves within the tube, it is found more advantageous to use an implicit algorithm such as the Newmark algorithm rather than an explicit algorithm. Actually, as the former is unconditionally stable (see for instance [AXI 04] Chapter 5), the time step of integration can be selected in relation to the characteristic time of the mass-spring response, independently of those of the natural modes retained in the basis, as appropriate. Computational results presented in the next two subsections were obtained by using a set of hundred acoustical modes. 6.2.3.1 Displacement field as the fluid variable The forced version of equations [6.24] is written as:
(K X s
s
+ M s X s + K L ( X s − X f
− ρ f ce2 S f
∂2 X f ∂x
2
))δ ( x ) = F ( ) (t )δ ( x ) e
+ ρ f S f X f + K L ( X f − X s ) δ ( x ) = 0
The system [6.111] is rewritten in dimensionless form as:
[6.111]
498
(κ X 2 0
μn
Fluid-structure interaction
s
∞ ⎛ f ⎞ + Xs + γ Lκ 02 ⎜ X s − ∑ X n( ) ⎟ = F m =0 ⎝ ⎠
)
(( nπ + α
out
)
2
(f)
Xn
(e)
[6.112]
)
⎛ ∞ ⎞ f f − Xn( ) + γ Lκ 02 ⎜ ∑ X n( ) − X s ⎟ = 0 n = 0,1, 2,… ⎝ m =0 ⎠
where the following dimensionless quantities are used: ξ=
x L
; τ=
K γL = L KS
tL ce
; Xs = X s / L 1
; μn =
⌠ μ f ⎮⎮ ⎮ ⌡0
( cos ( ( nπ + α
; X n(
out
f)
)ξ ))
2
= X n( ) / L f
dξ
; F
; κ0 = (e)
ω0 L ce
e F( ) ⎛ L ⎞ = ⎜ ⎟ LM s ⎝ ce ⎠
2
[6.113]
The scaling factors used in [6.113] to perform the computational work have been chosen to agree with those already introduced in subsection 6.2.1.2. However, to present the final results, other scaling factors can be preferred, which are found to be more pertinent in relation to the particularities of the problem of interest. The results presented here refer to the transient force signal depicted in Figure 6.14. As shown in the left-hand plot, it is shaped as a sinusoid truncated to a single period. The frequency f e = 10 Hz is equal to the natural frequency of the mass-spring system in vacuum and the peak amplitude is F0 = 10 N . The power spectral density of the force signal is depicted in the right-hand plot, using logarithmic scales. It is shaped as a rather broad and asymmetric bump peaking at 10 Hz.
Figure 6.14. Transient excitation: time history and power spectral density
Vibroacoustic coupling
499
Figure 6.15a. Open tube: time history and power spectral density of the piston displacement
In the same manner as for the excitation signal, the displacements of the piston is described using time history and power spectral density plots, see Figures 6.15a,b. In such plots, dimensionless time tr = t / T0 and frequency f r = f / f 0 are used, where T0 is the period and f 0 is the frequency of the mass-spring system in vacuum. The displacement is reduced by using the static response of the spring to the peak force F0 . The relative importance of the fluid to the solid properties is described again by the stiffness ratio γ f = K f / K s , where K f = E f S f / L and the mass ratio μ f = M f / M s . The plots of Figure 6.15a,b relate to the open tube case. Damping is
500
Fluid-structure interaction
assumed to be zero, and γ f is varied by modifying the speed of sound while the mass ratio is kept constant ( μ f = 2 ).
Figure 6.15b. Open tube: time history and power spectral density of the piston displacement
The results illustrated in Figures 6.15a,b can be briefly summarized as follows. The system behaves practically as a single degree of freedom system if γ f is either much larger or much smaller than one. In the former case, the fluid is practically incompressible in the frequency range of interest and the system behaves as an oscillator of stiffness coefficient K s and mass coefficient M e = M s + M f . In the second asymptotic case, presence of the fluid is negligible and the system behaves as an oscillator of stiffness coefficient K s and mass coefficient M e = M s , even if the mass ratio is larger than one. However, in a broad range of K f values, extending from a few percent to a few times K s , vibroacoustic coupling is clearly significant. In that range, the piston responds as a multiple DOF system. In qualitative agreement with the analytical results presented in subsections 6.2.2.1 to 6.2.2.3, if γ f is sufficiently small, the motion can be described as a series of damped vibrations modulated in amplitude by the periodical energy feedback at the large time scale τ of wave propagation. Thus, in practice, in the intermediate range of γ f values it would be difficult to anticipate the response of the system without
Vibroacoustic coupling
501
performing the detailed computation. As a rough approximation, the maximum displacement of the piston is varied from the in vacuum value to the incompressible fluid value. The magnitude of such a variation depends also in a rather intricate manner on the mass ratio μ f and on the power spectrum of the excitation signal.
Figure 6.16a. Closed tube: time history and power spectral density of the piston displacement
Therefore, even the maximum displacement of the piston cannot be easily anticipated without actually solving the coupled vibroacoustic problem. Such a conclusion is still reinforced when the case of a closed tube is analysed. As shown in the plots of Figures 6.16a,b which refer to a closed outlet, in the range γ f >> 1 the mass-spring system vibrates at a natural frequency much larger and an amplitude
502
Fluid-structure interaction
much smaller than in vacuum. In an intermediate range γ f 1 , the peak amplitude of the displacement is roughly the same as in the open tube case, though the spectral content of the response signals differ significantly. Finally, in the range γ f << 1 the response of the piston is almost independent of the boundary condition at the tube end and is the same as in vacuum.
Figure 6.16b. Closed tube: time history and power spectral density of the piston displacement
6.2.3.2 Response to a seismic excitation The response of the piston-fluid column system to a seismic excitation has been already described in Chapter 2, subsection 2.2.2.5, where the fluid was accounted for by its inertia only. As shown in Figure 6.17, the power spectral density of the acceleration signal prescribed to the spring is shaped as a broad bump extending essentially from about 1 Hz to 14 Hz, and peaked at about 7 Hz. The analysis is extended here to the case of a compressible fluid. Here again it is described in terms of the displacement field X f , as defined in the moving frame of reference.
Vibroacoustic coupling
503
Figure 6.17. Spectral content of the exciting acceleration signal
The forced version of the local equations of motion [6.24] is written in the time domain as:
(K X s
s
+ M s X s + K L ( X s − X f 2
− ρ f ce2 S f
∂ Xf
) ) δ ( x ) = − M D ( t ) δ ( x ) s
( t ) + ρ f S f X f + K L ( X f − X s ) δ ( x ) = − ρ f S f D
∂ x2
[6.114]
The equations [6.114], projected onto the modal basis, which includes the free oscillation of the mass-spring system in vacuum and the acoustical modes of the tube open at both ends, read as: ∞ ⎛ f ⎞ (τ ) κ 02X s + Xs + γ Lκ 02 ⎜ X s − ∑ X n( ) ⎟ = −τ 2 D m =0 ⎝ ⎠
μn
(( nπ + α
out
)
2
)
⎛ ∞ ⎞ f f f (τ ) X n( ) + Xn( ) + γ Lκ 02 ⎜ ∑ X n( ) − X s ⎟ = −τ 2 μ f β n D ⎝ m =0 ⎠
[6.115]
n = 0,1, 2,…
where the same dimensionless quantities as defined in [6.113] are used. Numerical integration of the system [6.115] is straightforward. A few illustrative results are presented in Figure 6.18. The physical system excited by the acceleration signal shown in Figure 6.17 corresponds to M s = 10 kg , f 0 = 10 Hz , μ f = 3 . The compressibility number κ 0 = ω0 L / ce is varied as a free parameter. The time histories of the piston response are presented as dimensionless plots using the reduced time tr = tL / ce and displacement X r = X s / D0 . Qualitatively, the general features of the response to a seismic excitation are similar to those already pointed out in the case of an external force, except in so far as the magnitude of the response
504
Fluid-structure interaction
is concerned in relation with the effective inertia of the coupled system, since the seismic forcing function is proportional to the effective inertia.
Figure 6.18. Time histories of the mass-spring displacement in dimensionless quantities
Starting from the upper left corner of Figure 6.18, the first plot corresponds to such a small value of κ 0 and acoustic travel time ( Ta / T0 0.003 ) that the fluid can be safely considered as practically incompressible. Accordingly, the free stage of the mass-spring response corresponds very accurately to that expected from the analytical calculation presented in Chapter 2, subsection 2.2.2.5, and could be obtained, though with a small error, by retaining in the modal basis the first acoustical mode only. The small error is attributed to the correction effect of the truncature stiffness coefficient. In the second plot, corresponding to κ 0 = 3 and Ta / T0 0.1 , the effect of compressibility is already noticeable. In particular, a significant decrease in the amplitude of the response is observed in comparison with the incompressible case. Another important point is that the free stage response is no longer monochromatic. The larger the κ 0 values, the more pronounced is the trend, see in particular the plot representative of the extreme case κ 0 = 1000 and Ta / T0 10 . Again, the response signal is marked by some kind of signature of beating involving the two time scales T0 and Ta = μ f T0 /(2πς va ) already identified
Vibroacoustic coupling
505
as a series of damped vibrations modulated in amplitude by the periodical energy feedback at the large time scale of wave propagation.
Figure 6.19. Time histories of the mass-spring displacement κ 0 = 104 , τ a 8 as computed by using modal bases of varying size
The case dominated by fluid compressibility gives us an opportunity to check the effect of the modal truncature on the computed response of the mass-spring system. Let us consider for instance the case μ f = 3 and κ 0 = 100 such that the number of acoustic modes having a frequency less than the upper bound f b 14 Hz of the excitation spectrum is about N c = 50. According to the spectral criterion, to obtain realistic results by using the modal method it is appropriate to include the N a = 50 acoustical modes into the modal basis. The time histories plotted in Figure 6.19 confirm that if N a is selected at a value substantially less than the appropriate cutoff value N c , the response is rapidly degraded. In particular, if the rigid mode n = 0 is the sole fluid mode retained in the basis, the computed response is found to identify essentially to that produced by the incompressible model. Such results are of practical importance as they point out that even if the fluid mass is correctly accounted for in the dynamical model, the computed dynamical response can be in large error if the modal basis is not appropriately truncated to account for the spectral content of the excitation signal.
506
Fluid-structure interaction
6.3. Vibroacoustic coupling in tube and ducts circuits 6.3.1
Simplifications inherent in the tubular geometry
As structural elements, tubes are usually described by using equivalent beam models. In practice several refinements of the basic beam theory can be necessary to account for various modes of deformation of the thin walled cross-sections, in particular ovalization of the cross-sections at the pipe elbows. However, here we shall restrict ourselves to the basic models analysed in depth in [AXI 05], which are sufficient for describing the fluid-structure coupling mechanism in pipes and ducts containing fluids. As a short preliminary, the major simplifying assumptions arising from the basic beam model are briefly recalled together with a few useful notations: 1. The characteristic length L of the pipe is assumed to be much larger than the transverse dimensions Dy and Dz . 2. As the normal cross-sections of the tube wall are supposed rigid, the displacement field used in the beam model comprises the small rotations about the centroid and the translations of the centroid. The former are described by the vector field ψ s ( s; t ) and the latter by the vector field X s ( s; t ) , where here s denotes the curvilinear abscissa along the tube axis. 3. The internal inertia and strain forces are described by using the mass and stiffness operators denoted M s ( s ) and K s ( s ) respectively, which are generally s dependent. It is stressed, if necessary, that both of them are formally self-adjoint and positive. When the pipe is filled with a fluid, the vibration modes of the equivalent beam are modified due to the combined effects of fluid inertia and compressibility, in agreement with the vibroacoustic coupling mechanism already introduced in section 2 of the present chapter. Furthermore, it is noted that the characteristic reduced wave numbers largely differ in the longitudinal and in the transverse directions, namely kDy , kDz << k L , in such a manner that the wavelengths involved in a vibroacoustic coupled mode are found to be generally much larger than the transverse dimensions of the tubes, except at frequencies higher than about ce / Dy , or ce / Dz , while they are similar or shorter than the length of the pipe. This peculiarity of the tubular geometry allows one to model the fluid-structure coupling differently according to the direction considered. The basic idea is to neglect the compressibility of the fluid, or not, depending whether transverse motions or longitudinal motions are concerned. Furthermore, the longitudinal acoustic fluctuations are treated by using the plane wave model already introduced in Chapter 4. Though essentially valid at a local scale, such an idea must however be refined when the pipes comprise bent parts where a coupling between the transverse and longitudinal effects take place, as explained later in subsection 6.3.2. This tubular vibroacoustic model is valid as long as the frequency range of interest
Vibroacoustic coupling
507
remains below the first transverse acoustic modes of the fluid contained in the pipe and below the first shell vibration modes of the pipe. Of course, this criterion can be restated in terms of transverse wavelengths. For instance, in the case of a circular cylindrical shell of radius R, based on the cut-off pulsation [5.135], the plane wave model is expected to be valid in the range: ω pw ωc
=
k pw kc
R <1 L
[6.116]
where ω pw stands for the characteristic pulsation, and k pw for the wave number of the plane acoustical wave, ωc and kc standing for the corresponding cut-off values below which only plane waves are not evanescent. Concerning the solid, the characteristic frequency which is relevant to the problem is related to the ovalization modes of the shell, which depend on the pipe geometry and support conditions. As a rough order of magnitude, it can be scaled by using the first ovalization mode of the ring equivalent to a shell strip of unit length. As shown in [AXI 05] formula [8.23], the corresponding natural frequency can be approximated as: ω2
cs ⎛ e ⎞ ⎜ ⎟ R ⎝R⎠
[6.117]
cs is the speed of sound in the solid and e the thickness of the shell. It is soon
realized that for thin walled tubes, the wave number k2 = ω2 / cs is likely to be significantly less than kc : ω2 ⎛ e ⎞ ⎛ cs ⎜ ⎟⎜ ωc ⎝ 2 R ⎠ ⎜⎝ c f
⎞ k e ⎟⎟ ⇔ 2 2 k R c ⎠
[6.118]
Actually, as further explained in what follows, even in the intermediate wave number range between ω2 and ωc , the tubular model can still be applied at the cost of some overestimation of the inertial effect, by an amount which remains insignificant in most applications. 6.3.2
Tubular vibroacoustic coupling model
The terms to be inserted into the vibration equations of the beam and the fluid column to model the vibroacoustic coupling can be derived formally by integrating the three-dimensional and linearized Euler equations on a tube strip in motion. Such a formulation is presented in [GIB 88]. However, as shown in this subsection, an elegant shorthand in the mathematics can be found by considering that, at the scale of an elementary strip of infinitesimal length dx, it is legitimate to treat the tube strip as a rigid solid filled with an incompressible fluid, which is clearly not the case at the global scale of the whole tubular circuit. Furthermore, it turns out that the coupling between the vibrations of the equivalent beam and the plane acoustical
508
Fluid-structure interaction
waves can be easily grasped and formulated when it is concentrated at particular places of the circuit, marked either by a sudden change in the cross-sectional area of the fluid column or by a sudden change in the direction of the tube axis. Generalization of the discrete formulation to the continuous case, to deal with progressive changes in the geometry of the fluid column, is straightforward. 6.3.2.1 Incompressible transverse coupling terms
Figure 6.20. Elementary strip of the tube supported by springs acting in translation
Consider an elementary strip of a circular cylindrical tube of internal radius R and infinitesimal length dx supported by springs acting in translation, as depicted in Figure 6.20. M s is the mass of the solid and M f that of the internal fluid. The displacement of the strip is entirely described by the translation field written in the local Cartesian coordinate system of unit vectors t , n1 , n2 as: X s ( s, t ) = X s t + Z s n2 [6.119] We note that a longitudinal translation X s t of the pipe induces no motion in the fluid, provided fluid viscosity is discarded, and reciprocally the fluid exerts no fluctuating force on the tube walls. That is, the fluid column and the tube are entirely decoupled for this type of motion. Of course, some coupling occurs for a real fluid, as a consequence of the tangential viscous forces at the tube walls. This aspect of the problem will be analysed in Chapter 7 which deals with dissipative effects. In contrast with the longitudinal case, in a transverse translation, Z s n2 for instance, the fluid is clearly constrained to follow the tube. Furthermore, if the motion is slow enough so that ω R / ce << 1 , fluid compressibility can be neglected and the fluid moves as a rigid solid. The forces exerted on the tube walls are inertial in nature and their resultant turns out to be simply F f = − M f Zs n2 . Actually,
Vibroacoustic coupling
509
mathematical proof of this intuitive result is found in Chapter 2, subsection 2.3.2.1, where it was presented in the context of a circular cylindrical shell of infinite length (strip model) filled with an incompressible fluid. Indeed, the transverse translation of the tube element corresponds to the n = 1 shell mode and the added mass coefficient per unit shell length derived from formula [2. 125] was found to be precisely: ma ( n ) =
ρ f π R2 n
[6.120]
Accordingly, the inertial effect of the internal fluid on the transverse motion of the tube can be very easily accounted for by adding the two-dimensional added mass coefficient ma (1) to the cross-sectional mass of the tube, as far as the transverse equations of the tube motion (displacement fields Y ( x; t ) and Z ( x; t ) normal to the tube axis) are concerned. So, for the rigid tube shown in Figure 6.20, the free oscillations are governed by the longitudinal and transverse equations: ⎡⎣ K x − ω 2 M s ⎤⎦ X s = 0
⎡ K z − ω 2 ( M s + M f )⎤ Z s = 0 ⎣ ⎦
[6.121]
which are found appropriate to point out the anisotropic nature of the fluid-structure coupling.
Figure 6.21. Pair of tubes coupled by fluid inertia: (a) coaxial configuration, (b) tandem configuration
On the other hand, the inertial coupling associated with tube ovalization can be accounted for in the same way, by using the appropriate added mass coefficient ma ( 2 ) instead of ma (1) . Of practical importance in many applications is the extension of the procedure to the inertial effects induced by an external fluid. As already analysed in Chapter 2, subsection 2.3.2.1, if the tube is immersed in an infinite extent of fluid, the added mass coefficient per unit tube length is:
510
Fluid-structure interaction
ma = π ρ i Ri2 + ρ e Re2
[6.122]
where ρ i and ρ e denote the density of the internal and external fluid, the internal and external radius of the tube wall being denoted Ri and Re respectively. In other configurations of practical importance, several tubes can either be nested coaxially or closely packed as a tube array. As illustrated in Figure 6.21 for a pair of tubes, the interstitial fluid couples the transverse displacements of the tube by inertia forces, which can be computed by using a strip model as long as the slenderness ratio of the tube is sufficiently large. The case of two coaxial tubes is particularly simple since the inertial matrix is easily derived analytically as already seen in Chapter 2, subsection 2.3.2.3. Formula [2.156] particularized to the case n = 1, is rewritten here for convenience as: ⎡ R12 ( R12 + R22 ) −2 ( R1 R2 )2 ⎤ ⎥ [Ma ] = 2 2 ⎢ R2 − R1 ⎢ −2 ( R R )2 R 2 ( R 2 + R 2 ) ⎥ 1 2 2 1 2 ⎦ ⎣ ρ fπ
[6.123]
R 1 is the external radius of the internal tube and R 2 the internal radius of the
external tube. In the case of non coaxial tubes and tube arrays, the mechanism of transverse coupling remains of course the same; the coefficients of the coupling matrix are however much less easily determined than in the coaxial case, requiring generally numerical computations based on the finite element method. To conclude on the inertial coupling effects, it is noted that any coupling between the rotation field of the pipe ψ s ( s; t ) and fluid inertia is neglected; in the same way as, in the Bernoulli-Euler model, the inertia of the cross-sections related to bending rotation is discarded. 6.3.2.2 Vibroacoustic coupling at a change in the cross-section Plane sound waves and longitudinal tube vibration are coupled to each other at every place where the cross-sectional area of the fluid column varies. The simplest way to formulate this coupling mechanism is to consider first the discrete case of an abrupt change in the cross section Δ S = S2 − S1 concentrated at some curvilinear abscissa s0 along the tube axis. As shown in Figure 6.22, in the neighbourhood of the geometrical singularity, the pipe can be modelled as two straight and coaxial tube elements of infinitesimal length ε , which contain two fluid columns of crosssectional area S1 and S2 respectively. The junction at the change of cross-section is provided by a rigid annular plate normal to the tube axis of area Δ S .
Vibroacoustic coupling
511
Figure 6.22. Vibroacoustic coupling at a sudden change in the cross-sectional area of a tube
Since ε is supposed much smaller than both the fluid sound and tube vibration wavelengths, the displacement field of the tube can be considered as uniform X s ( so ; t ) , at the local scale s0 − ε ≤ s ≤ s0 + ε at least, and so the plane wave sound pressure field p ( so ; t ) . The longitudinal motion of the junction plate induces a concentrated volume velocity source of plane sound waves, which is simply: Q ( so , t ) = ( S2 − S1 ) X s ( xo ; t ) . = ( S2 − S1 ) X s [6.124] The fluctuating pressure field induces the resulting force on the rigid plate: F ( so ; t ) = − p ( so ; t ) ( S2 − S1 )
[6.125]
The results are easily transposed from the discrete to the continuous case by using a differential formulation instead of finite discontinuities. Considering an elementary strip of tube of variable cross-sectional area S(s), the source density of volume velocity exerted on the fluid per unit tube length is: dQ dS =+ X s ( s; t ) . ( s ) [6.126] ds ds The density of force exerted on the wall per unit tube length is: ∂F dS = − p ( s; t ) (s) ∂s ds
[6.127]
512
Fluid-structure interaction
At this step, it must be stressed that no vibroacoustic coupling terms are related to the rotation field of the tube ψ s ( s; t ) . This because an angular motion of the wall does not induce any change in the volume offered to the fluid, and therefore no fluctuating pressure. Furthermore, as long as viscosity of the fluid is discarded, no tangential stresses arise. 6.3.2.3 Vibroacoustic coupling at bends Plane sound waves and tube vibration are also coupled to each other at every place where there is a change in the direction of the tube axis, which is typically the case in curved and mitred tubes. Here also, the simplest way to formulate the coupling terms is to consider first the discrete case of a mitre bend of angle θ0 , as shown in Figure 6.23. Again in the neighbourhood of the geometrical singularity, concentrated at s0 , the pipe can be modelled as two straight tube elements of infinitesimal length ε and axial unit vectors 1 and 2 . The cross-sectional area is constant. Within the interval s0 − ε ≤ s ≤ s0 + ε , the displacement field of the tube and the fluctuating pressure field can be assumed uniform, equal respectively to D ( so ; t ) and p ( so ; t ) . Elegant shorthand to determine both the resultant of the pressure force exerted on the wall of the pipe at s0 and the source of volume velocity induced by the pipe motion consists in closing mentally the inlet and the outlet of the mitre using rigid terminations as shown in Figure 6.23. Since pressure is uniform, the resulting pressure force R exerted on the walls of the closed volume is nil. Such a result, which agrees with the common experience concerning inflated structures of various shapes, like balloons, tyres etc. can be also proved in a more formal way using the force balance: ⌠ R = ⎮⎮
⌡( Σ )
⌠ pnd Σ = ⎮⎮ grad p.ndV ⌡(V )
[6.128]
where ( Σ ) is the surface bounding the volume (V ) of the mitre closed at both ends. As the pressure is essentially uniform at the length scale ε , the pressure gradient is negligible and so also the resulting force R . On the other hand, since the fluid is practically incompressible at the length scale ε and since the motion of the mitre is assimilated to a uniform translation, ( Σ ) and (V ) remain unchanged and the flow across ( Σ ) is nil, as a consequence of the mass conservation law: ⌠ ⌠ Q = ρ f ⎮⎮ V f .nd Σ = ⎮⎮ ⌡( Σ )
⌡(V )
ρ f div V f dV = 0
[6.129]
Starting from the results [6.127] and [6.129], the fluid-structure coupling terms induced at the walls of the mitre in the actual configuration (open ends) can be
Vibroacoustic coupling
513
Figure 6.23. Sudden change in the direction of the tube axis, or mitre bends, Fp ( t ) and
Q ( t ) as defined in the figure stand for the vibroacoustic coupling terms
immediately deduced from the corresponding terms which would occur at the closed ends. Using such a procedure, the source of volume velocity induced by the tube motion is found to be the negative of the volume velocity through the end cross sections: Q ( so ; t ) = − S 2 − 1 .D ( so ; t ) [6.130]
(
)
In the same way, the force exerted on the walls by the fluctuating pressure is the negative of that exerted at the end cross-sections: F ( so ; t ) = − p ( so ; t ) S 2 − 1 [6.131]
(
)
Figure 6.24. Smooth bend of curvature radius Rc
The results [6.130] and [6.131] are easily transposed from the mitre bend to the smooth bend shown in Figure 6.24 as follows:
514
Fluid-structure interaction
dQ ( s; t ) = ds dF ( s; t ) = ds
S D.n Rc
[6.132]
pS n Rc
[6.133]
where the unit tangent vector ( s ) to the tube axis points toward increasing s values. The unit normal vector n ( s ) is oriented in such a manner that the two
following equations are satisfied: d n dn ; =− = ds Rc ds Rc
[6.134]
As shown in Figure 6.24, Rc is the positive radius of curvature, C(s) is the curvature centre and n points from C(s) to M(s); in other words, it is oriented toward the extrados of the bend. 6.3.2.4 Vibroacoustic coupling at closed ends and tube junctions The coupling terms formulated in the two preceding subsections can be immediately extended to the particular cases of piping branches terminated by a closed end. Of course such singularities can be treated as abrupt changes in the cross-sectional area of the fluid column. However, it is also necessary to distinguish between a sudden enlargement Δ S = + S and a sudden narrowing Δ S = − S . This is easily achieved by adopting, as a convention, a positive direction along the network as if it would convey a steady flow. Then, a sudden enlargement corresponds to a “upstream termination” and a sudden narrowing to a “downstream” termination, as indicated in Figure 6.25. Moreover, if several branches of the pipework are connected through a (small) junction element (VJ ) , whose dimensions are much less than the wavelengths of interest, the corresponding coupling terms can be easily derived by using the same procedure as in the case of a bend. Indeed, as long as VJ is small enough, the displacement field of the junction element and the fluctuating pressure field in the enclosed fluid can be assumed uniform.
b g
As a consequence, if (VJ ) is closed by plates S1 , S2 ,..., S N at the outlets or inlets of the connected branches, the resulting volume velocity and pressure force are found to be nil. Therefore, the conditions to be fulfilled in the actual configuration are as follows: pn = pJ ( so ; t ) n = 1,..., N
Qn = + Sn DJ ( so , t ) . n Fn = + pJ Sn n
[6.135] [6.136] [6.137]
Vibroacoustic coupling
515
Figure 6.25. Rigid tube elements: closed ends and junctions
Equations [6.136] to [6.137] formulate the coupling terms to apply at the inlet, or at the outlet of each branch to ensure the mechanical equilibrium and fluctuating flow continuity at the connecting element VJ . Hence, such terms can be accounted for by using plates S1 , S2 ,..., SN with the same sign with respect to the upstream or downstream type of termination which would be used to close VJ .
b g
b g
6.3.2.5 Equation of motion of a pipe filled with a fluid The partial results established in the last four subsections can be suitably gathered together to formulate the vibroacoustic coupled equations for a curved pipe with varying cross-section filled with a compressible fluid, see Figure 6.26. The beamlike vibration of the pipe is governed by the following equation of motion: dS e pS K s ⎡⎣ X s ⎤⎦ + M s ⎡⎢ X s ⎤⎥ + ρ f S f X s .t t − f n + p f = F ( ) ( s; t ) ⎣ ⎦ Rc ds
( )
Ks [
]
and M s [
]
[6.138]
are the stiffness and mass operators of the equivalent beam
element. The first fluid term in the left-hand side of [6.138] accounts for the inertia force of the fluid which is constrained to follow the motion of the tube in any transverse direction t . As the field displacement X s does not include any shell flexural modes, the added mass coefficient per unit tube length is ρ f S f . The two other fluid terms account for the vibroacoustic coupling forces exerted by the plane sound waves on the tube walls at any place where there is a change either in the tube direction, or the fluid cross-section. The source term in the right-hand side of the equation is the external fluctuating load exciting the equivalent beam.
516
Fluid-structure interaction
Figure 6.26. Curved pipe of non uniform cross-section
The fluctuating pressure field in the fluid column is governed by the following plane sound wave equation: dS f X s .n ∂ ⎛ ∂ p ⎞ Sf ∂ 2 p S ρ S X s . = S ( e ) ( s; t ) − − +ρf [6.139] ⎜ f ⎟ f f 2 2 ∂ s ⎝ ∂ s ⎠ ce ∂ t Rc ds The two solid terms in the left-hand side of [6.139] stand for the vibroacoustic volume velocity sources induced in the contained fluid by the beamlike motion of the tube. Though the field displacement X s does not include any shell breathing modes, dilatation of the tube cross-sections is accounted for by replacing the speed of sound in the infinite fluid c f by the equivalent value ce calculated by using the relation [4.277], or [4.283] in the case of a circular cylindrical tube. The source term in the right-hand side of the wave equation is the external fluctuating load exciting the plane sound waves. On the other hand, the coupled equations [6.138] and [6.139] can be transformed into a symmetrical system by using the pressure and Π variables to describe the fluid, as introduced in subsection 6.1.3. The coupled beam equation [6.138] is rewritten as: ⎛ dS f S f n ⎞ (e) ⎡ ⎤ − K s ⎣⎡ X s ⎦⎤ + M s ⎢ X s ⎥ + ρ f S f X s .t t + ⎜ [6.140] ⎟ Π = FS ( s; t ) ⎣ ⎦ Rc ⎠ ⎝ ds
{
( )}
The coupled fluid column equation becomes: ⎧ dS f S f ⎫ 1 − n ⎬.X s + ⎨ Rc ⎭ ρf ⎩ ds
⎞ S f ⎫⎪ 1 ( e ) ⎧⎪ ∂ ⎛ ∂ Π p⎬ = S ( s; t ) ⎨ ⎜Sf ⎟− ∂ s ⎠ ce2 ⎭⎪ ρ f ⎩⎪ ∂ s ⎝
[6.141]
As desired, the system of equations [6.140], [6.141] and [6.6] takes on the canonical form:
Vibroacoustic coupling
e K va′′ ( X va ) + M va′′ ( X vas ) = Fva( ) ( s; t )
517
[6.142]
where the vibroacoustic operators are defined as follows: ⎡ dS f S f n − ⎢ M s + ⎡⎣ ρ f S f ⎤⎦ t ⎡ ⎤ ds Rc ⎢ ⎢K 0 0 ⎥ ⎢ dS S n ⎢ s ⎥ ∂ ⎞ 1 ∂ ⎛ f K va′′ = ⎢ 0 0 0 ⎥ ; M va′′ = ⎢ Sf − f ⎜ ⎟ ⎢ ds Rc ρf ∂ s ⎝ ∂ s⎠ ⎢ ⎥ S ⎢ f ⎢0 0 ⎥ −S f ⎢ ⎢⎣ ce2 ⎥⎦ 0 ⎢ ρ f ce2 ⎣
⎤ 0 ⎥ ⎥ −S f ⎥ ⎥ ρ f ce2 ⎥ ⎥ 0 ⎥⎥ ⎦
[6.143]
The vibroacoustic mixed displacement vector and external load are written as follows: T ⎡⎣ X va ⎤⎦ = ⎡⎣ X s
Π
e T ⎡e p ⎤⎦ ; ⎡⎣ Fva( ) ⎤⎦ = ⎢ Fs( ) ( s; t ) ⎢⎣
S ( ) ( s; t ) e
ρf
⎤ 0⎥ ⎥⎦
[6.144]
K va′′ and M va′′ are the vibroacoustic mixed stiffness and mass operators, respectively. Once more, it is stressed that they are formally self-adjoint, K va′′ is nil and M va′′ is positive definite. The free harmonic oscillations of the system are governed by the matrix equation: ⎡ ⎡ ⎤⎤ dS f S f n ⎡ ⎤ + ρ − M S 0 ⎥⎥ ⎢⎡ ⎢ s ⎣ f f ⎦t ⎤ ds Rc ⎢⎢ ⎢ ⎥⎥ ⎥ 0 ⎥ ⎢⎢Ks 0 ⎢ dS S n ⎥ ⎥ ⎡ X s ⎤ ⎡0⎤ − S ∂ ∂ 1 ⎛ ⎞ f f f 2 ⎢⎢ 0 0 ⎥ ⎥ ⎢ Π ⎥ = ⎢0⎥ − 0 ⎥ −ω ⎢ ⎜Sf ⎟ ⎢⎢ ⎢ ds Rc ρ f ∂ s ⎝ ∂ s ⎠ ρ f ce2 ⎥ ⎥ ⎢ ⎥ ⎢ ⎥ ⎥ Sf ⎥ ⎢⎢ ⎢ ⎥ ⎥ ⎢⎣ p ⎥⎦ ⎢⎣0⎥⎦ −S f ⎢ ⎢ 0 0 ρ c2 ⎥ ⎥ ⎢ ⎥ 0 0 ⎥⎥ f e ⎦ 2 ⎢⎣ ⎢ c ρ f e ⎣ ⎦⎦ ⎣ [6.145]
As a consequence of the nullity of K va′′ , there are infinitely many non trivial solutions of the kind: ω = 0 ; Xs = 0 ;
p=0 ; Π ≠0
[6.146]
Physically, they stand for steady fluid motions about static structures. 6.4. Application to a few problems The object of this subsection is to solidify the general formalism just described above by solving a few problems especially selected to illustrate some important points concerning vibroacoustic coupling in pipe systems.
518
Fluid-structure interaction
6.4.1
Vibroacoustic modes of cylindrical vessels
6.4.1.1 Longitudinal vibroacoustic modes of a straight vessel As a first exercise, we analyse here the vibroacoustic modes of the cylindrical vessel shown in Figure 6.27, which is modelled as a straight tube closed at both ends and containing a fluid modelled as a straight fluid column of uniform cross-sectional area. Finally K 0 is the stiffness coefficient of the axial supports of the vessel, assumed to be much less than the characteristic stiffness of the beam: K 0 <<
Es Ss L
[6.147]
Figure 6.27. Elongated cylindrical vessel
To solve the problem semi-analytically by using the modal projection method, it is first chosen to formulate the motion of both the structure and the fluid, in terms of displacement. In agreement with equations [6.24], the coupled system is formulated as: − ρ s cs2 S s 2 e
−ρ f c S f
d2Xs − ω 2 ρ s Ss X s + K0 X s + K L ( X s − X f d x2
(
d2X f d x2
)) δ ( x ) = 0 [6.148]
− ω ρ f S f X f + KL ( X f − X s )δ ( x ) = 0 2
Here, the hemispherical caps closing the vessel have been discarded in the beam equation, though it would be straightforward to include them in the model as additional masses concentrated at the beam ends. Discretization of the system [6.148] is performed by using the longitudinal modes of the non supported beam and fluid column:
Vibroacoustic coupling N ⎛ nπ x ⎞ X s = ∑ an cos ⎜ ⎟ ⎝ L ⎠ n =0
N ⎛ nπ x ⎞ X f = ∑ bn cos ⎜ ⎟ ⎝ L ⎠ n =0
;
Modal projection is straightforward, producing a 2 N × 2 N equation of the type:
519
[6.149] algebraic matrix
⎡ ⎡ ⎡[ K ] + ⎡ K ⎤ ⎤ ⎤ − ⎡⎣ K sf ⎤⎦ ⎡[ M ] [0] ⎤ ⎤⎥ ⎡[ a ]⎤ ⎡[0]⎤ ⎢ ⎢ ⎣ ss ⎣ sf ⎦ ⎦ ⎥ − ω 2 ⎢ ss ⎥ ⎢ ⎥=⎢ ⎥ T ⎢⎢ ⎡ ⎤⎥ ⎢⎣ [ 0] ⎡⎣ M ff ⎤⎦ ⎥⎦ ⎥⎥ ⎣[ b] ⎦ ⎣[0]⎦ ⎢⎣ ⎣⎢ − ⎣⎡ K sf ⎦⎤ ⎣ ⎣⎡ K ff ⎦⎤ + ⎣⎡ K sf ⎦⎤ ⎦ ⎦⎥ ⎦
[6.150]
where the coefficients of the matrices are as follows: E S ( nπ ) K ss ( n, n ) = s s + K0 2L 2
E f S f ( nπ )
; K ss ( n, m ) = 0 if n ≠ m
2
; K ff ( n, m ) = 0 if n ≠ m 2L ρS L ; M ss (1,1) = ρ S S S L ; M SS ( n, m ) = 0 if n ≠ m M ss ( n > 1, n > 1) = s S 2 ρ S L ; M ff (1,1) = ρ f S f L ; M ff ( n, m ) = 0 if n ≠ m M ff ( n > 1, n > 1) = f f 2 K ff ( n, n ) =
(
K sf ( n, m ) = K L 1 + ( −1)
n+m
)
[6.151] As could be anticipated based on the results already established in section 6.2, the importance of vibroacoustic coupling is controlled by the fluid to solid stiffness and mass ratios: γf =
Ef Sf Es S s
; μf =
Mf Ms
[6.152]
The main features of the longitudinal vibroacoustic modes are best illustrated by considering first an intermediate case, as illustrated in Table 6.1 and Figure 6.28 which present results obtained by solving the system [6.150] with the following numerical parameters: Geometry: L = 2 m ; R = 10 cm ; e = 3mm Steel: Es = 21011 Pa ; ρ s = 8103 kg/m 3 ; ρ s S s 15.3kg/m ; cs = 5000m/s Water: E f = 2.3109 Pa ; ρ f = 1000kg/m3 ; ρ f S f 31.4kg/m ; c f = 1500m/s
520
Fluid-structure interaction
Figure 6.28. Coupled mode shapes: solid in full line and fluid in dashed line
Vibroacoustic coupling
521
The effective speed of sound in the vessel ce = 1134m/s is substantially less than in infinite medium. Such values lead to the solid stiffness and mass ratios γ f = 0.18 and μ f = 2.05 . The natural frequencies of the unsupported vessel in vacuum are f s( ) ≅ 1250n Hz n
and those of the fluid column are f f( n ) ≅ 284n Hz , n = 0,1, 2,... Starting from such values, the stiffness coefficient of the support has been selected in such a way that the first vibroacoustic mode is expected at 100 Hz, a frequency at which fluid compressibility is practically negligible. Table 6.1 lists the computed natural frequencies of the first twelve modes. The frequency values reported in the left-hand side column are relative to the uncoupled system ( ⎡⎣ K sf ⎤⎦ = [ 0] ). The acoustical or structural nature of the uncoupled mode is specified in the central column, by the symbol (A) and (S) respectively. In the frequency interval explored, there are nine fluid modes and only three solid modes. The fluid modes comprise the uniform fluid displacement mode at zero frequency, and a harmonic series of eight acoustical modes, with the fundamental at 284 Hz. The first s
structural mode is the longitudinal rigid mode at f 0( ) =
(
)
K 0 / M s / 2π = 175 Hz .
The two following modes stand for the fundamental and the first harmonic of the non supported beam. Finally, the frequency values reported in the right-hand side column are related to the coupled system. As could be expected, the frequencies of the vibroacoustic modes differ substantially from those of the uncoupled system. Furthermore, it is of interest to quote the values which would be produced by a simplistic calculation in which compressibility of the fluid would be discarded. In the frequency interval explored, only three structural modes would exist, at 100, 2023 and 3035 Hz. The practical importance of including fluid compressibility in the model is further illustrated in Figure 6.28 which displays a sample of coupled mode shapes. The displacement field of the solid is plotted in full line and the dashed line corresponds to that of the fluid. The plot in the upper left corner indicates clearly that the computed frequency is somewhat less than the value expected for an incompressible fluid (96.3 instead of 100 Hz) simply because at about 100 Hz compressibility is not totally negligible. For all the other modes, motion of the structure differs markedly from that of the fluid, invalidating thus the incompressible fluid model. As the structure is significantly more rigid than the fluid, the vibroacoustic mode shapes combine in different ways a same type of structural motion with distinct fluid modes. For instance, the modes indexed 1,3,6 correspond to a displacement field of the structure with a single antinode. They differ qualitatively from each other by the number of antinodes present in the displacement field of the fluid, which are 1,3,5 respectively. A similar pattern can be recognized concerning the modes 2,4,5,7. On the other hand, for the 1,3,6 modes the fluid and solid displacement fields are symmetrical about the mid cross-section whereas for the 2,4,5,7 modes they are antisymmetric.
522
Fluid-structure interaction Table 6.1. Closed tube: longitudinal vibroacoustic modes
Uncoupled modes Frequency (Hz)
Nature of the mode
Coupled modes Frequency (Hz)
0
(A)
96.5
175
(S)
424
284
(A)
509
567
(A)
897
850
(A)
972
1250
(S)
1364
1417
(A)
1409
1700
(A)
1800
1984
(A)
1908
2268
(A)
2300
2500
(S)
2343
2551
(A)
3735
In the range γ f <<1 and μ f <<1 , vibroacoustic coupling is negligible and the modes of the system reduce to those of the solid in vacuum and to those of the fluid column closed at both ends. The domain γ f >>1 and μ f >>1 is more interesting from the theoretical than from the practical standpoint, as it is not easily achieved in practice. Actually, it would require a high degree of anisotropy in the elastic behaviour for the tube, which must be highly flexible in the longitudinal direction and still sufficiently stiff in the radial direction to avoid a drastic lowering of the speed of sound ce in the fluid column in comparison with the corresponding value c f in infinite medium. Assuming that such a condition can be fulfilled, for the low frequency modes of the structure the fluid is practically incompressible in such a way that the structural modes are constrained to comply with the condition of equivolumic motion, which implies: L
⌠ ⎮ ⎮ ⎮ ⎮ ⌡0
∂ϕ n( ∂x
X)
dx = ϕ n(
X)
( L ) − ϕ n( X ) ( 0 ) = 0
[6.153]
Vibroacoustic coupling
523
Figure 6.29. Cylindrical vessel structural modes: uncoupled calculation, γ f = 2.6104 ,
μ f = 65 . The dashed line helps to locate the displacement nodes
As a consequence, the modal features are significantly changed, as the reader can verify by comparing Figures 6.29 and 6.30 which refer to the following numerical parameters: Geometry: L = 2 m ; R = 10 cm ; e = 1cm Soft rubber: Es = 5106 Pa ; ρ s = 950 kg/m 3 ; cs = 72.5 m/s Mercury: E f = 2.81010 Pa ; ρ f = 1.3104 kg/m 3 ; c f = 1450 m/s ce is artificially supposed to be equal to c f , so γ f = 2.6104 and μ f = 65 . The
difference between the results of the uncoupled and coupled calculations is obvious, as fluid-structure coupling prevents any structural displacement at the tube ends. Furthermore, no fluid motion is associated with these structural modes. Hence, the natural frequencies remain essentially unchanged from one calculation to the other, except of course as the first mode of rank zero is concerned, In the non coupled calculation it stands for the constant pressure mode at zero frequency and in the coupled calculation it becomes a constant pressure mode associated with an equivolumic tube displacement at low (but not zero) frequency. Finally, study of the asymptotic case γ f <<1 and μ f >>1 is left to the reader as an exercise.
524
Fluid-structure interaction
Figure 6.30. Cylindrical vessel structural modes: coupled calculation, γ f = 2.6104 ,
μ f = 65 . The dashed line helps to locate the displacement nodes
6.4.1.2 Numerical aspects related to the modal projection method The exercise of the last subsection gives us a good opportunity to discuss a few interesting aspects concerning the substructuring method used together with the modal projection method. At first, the use of the stiffness coefficient K L as a penalty factor to link the solid and the fluid together can be suitably elucidated by projecting the system [6.148] onto a modal basis restricted to the solid and fluid modes of index n = 0. Because in such modes the solid and the fluid behave as two rigid bodies, rigidly connected to each other, in principle at least, the ideal expected result is the coupled mode at 100 Hz related to a uniform displacement identical for the fluid and the structure. Actually, the modal matrix equation [6.150] restricted to the modes n = 0 , is: ⎡ ⎡ K0 + K L ⎢⎢ ⎢⎣ ⎣ − K L
−KL ⎤ ⎡M s − ω2 ⎢ ⎥ KL ⎦ ⎣ 0
0 ⎤ ⎤ ⎡ a0 ⎤ ⎡[0]⎤ = ⎥ M f ⎥⎦ ⎥⎦ ⎢⎣ b0 ⎥⎦ ⎢⎣[0]⎥⎦
[6.154]
The characteristic polynomial reads as:
(
ω 4 M s M f − ω 2 K0 M f + K L ( M s + M f
)) + K K 0
L
=0
[6.155]
Vibroacoustic coupling
525
This reduces practically to the desired equation, −ω 2 ( M s + M f ) + K 0 = 0
[6.156]
provided K L is sufficiently large to fulfil the condition: MsM f K0 K L
=
1 << 1 ω ωL2
[6.157]
2 0
Then, if that is the case, the second row of equation [6.154] reduces practically to: K L ( a0 − b0 ) = 0 ⇒ a0 = b0
[6.158]
which is the desired result concerning the mode shapes. The second mode of the model is a high frequency out-of-phase mode where the solid and the fluid vibrate in phase opposition at a frequency which depends on the value selected for the stiffness coefficient K L of the connecting spring. Of course, such a mode is merely a numerical artefact and is physically meaningless. The second point worth to be discussed is the mixed formulation of the problem using pressure instead of displacement as the fluid variable. In agreement with equations [6.45], the coupled system is governed here by the following equations: ∂2 X s + K 0 X sδ ( x ) − ω 2 ρ s S s X s + S f ( δ ( x ) − δ ( x − L ) ) p = 0 ∂x 2 d 2 p ω2 + p − ω 2 ρ f (δ ( x ) − δ ( x − L ) ) X s = 0 dx 2 ce2
− Es Ss
[6.159]
As in subsection 6.2.1.3, it is found convenient to rewrite the system [6.159] in a dimensionless form by using the following scaling factors: ξ=
x L
; Xs =
K L γ0 = 0 Es S s −
Xs p p = ; ϖ= 2 ρ f ce E f L
; γf =
Ef Sf Es S s
c2 ; χ = e2 cs
; κ=
ωL ce
; μf =
Mf Ms
d 2X s + γ 0X sδ (ξ ) − χκ 2X s + S f ϖ (δ (ξ ) − δ (ξ − 1) ) = 0 dξ 2
d 2ϖ + κϖ − κ 2X s (δ (ξ ) − δ (ξ − 1) ) = 0 dξ 2
[6.160]
[6.161]
The system [6.161] is discretized by using the same pressure mode shapes as in subsection 6.2.1.3. Hence, the motion of the vessel is described in terms of the longitudinal modes of the unsupported structure whereas the motion of the contained fluid is described in terms of the acoustic modes of the fluid column stopped at both ends. Accordingly, the displacement and pressure fields are expanded as follows:
526
Fluid-structure interaction N
N
S ⎛ nπ x ⎞ X s = ∑ an cos ⎜ ⎟ ⎝ L ⎠ n =0
f ⎛ nπ x ⎞ ; ϖ = ∑ bn cos ⎜ ⎟ ⎝ L ⎠ n =0
[6.162]
Projection of [6.161] onto the basis [6.162] produces a
(N
fS
+ 2 ) × ( N fS + 2 ) non
symmetrical algebraic system, where N fs = N f + N s , of the kind: ⎡ ⎡[ K ' ss ] ⎢⎢ ⎢ ⎢ [ 0] ⎣⎣
[ K ss′ ]
⎡ [ M ' ss ] [0] ⎤ ⎤ ⎡[ a ]⎤ ⎡[0]⎤ ⎣⎡ K ' sf ⎦⎤ ⎥⎤ −κ 2 ⎢ ⎥⎥ ⎢ ⎥ = ⎢ ⎥ ⎡⎣ K ' ff ⎤⎦ ⎥⎦ ⎢⎣ ⎡⎣ M ' sf ⎤⎦ ⎡⎣ M ' ff ⎤⎦ ⎥⎦ ⎥⎦ ⎣[ b] ⎦ ⎣[0]⎦
and
[ M ss′ ] stand
for the
( N s + 1) × ( N s + 1)
[6.163]
structural stiffness and mass
matrices, respectively which are the dimensionless version of [6.150]: K ss′ ( n, n ) =
1 2 ( nπ ) + γ 0 2
; K ss′ ( n, m ) = 0 if n ≠ m
1 ; M ss′ ( n > 1, n > 1) = χ 2
M ss′ (1,1) = χ
[6.164] ; M ss′ ( n, m ) = 0 if n ≠ m
⎡⎣ K ′ff ⎤⎦ and ⎡⎣ M ′ff ⎤⎦ stand for the ( N f + 1) × ( N f + 1) fluid stiffness and mass dual matrices, respectively. Both are diagonal, the coefficients are found to be: K ′ff ( n, n ) = −
( nπ )
2
2
( n = 0,1, 2,..., N ) f
M ff (1,1) = −1 ; M ′ff ( n > 1, n > 1) = −
[6.165]
1 2
Finally, the fluid-structure coupling terms are described by the rectangular matrices ⎡⎣ K sf′ ⎤⎦ and ⎡⎣ M sf′ ⎤⎦ . Both are full and they differ from each other by a multiplying factor and a transposition only. ⎡⎣ K sf′ ⎤⎦ is a ( N s + 1) × ( N f + 1) matrix whose coefficients are:
(
K sf′ ( n, m ) = γ 1 − ( −1)
n+m
)
n = 0,1,… N S
; m = 0,1,… N f
[6.166]
⎡⎣ M sf′ ⎤⎦ is a ( N f + 1) × ( N s + 1) matrix whose coefficients are:
(
M sf′ ( m, n ) = 1 − ( −1)
n +m
)
n = 0,1,… N S
; m = 0,1,… N f
[6.167]
A few mode shapes including the pressure profiles in the fluid (dashed line) and the displacement fields of the structure (full line) are plotted in Figure 6.31. Furthermore, the displacement field of the structure has been artificially multiplied by a factor ten to help visualize the displacement and pressure components in a same plot. The natural frequencies of the first thirteen modes are listed in Table 6.2.
Vibroacoustic coupling
527
Figure 6.31. Coupled mode shapes: solid displacement and fluid pressure field (dashed lines)
528
Fluid-structure interaction Table 6.2. Numerical approximations of the natural frequencies
Displ. formulation (Xs ; X f )
Mixed formulation ( X s ; p)
N s = N f = 10
N s = N f = 10
Frequency (Hz)
Frequency (Hz)
Displ. formulation (Xs ; X f ) N s = N f = 50
Frequency (Hz)
Mixed formulation ( X s ; p) N s = N f = 50
Frequency (Hz)
—
0
—
0
97
97
96.3
96.5
430
430
424
424
524
511
509
508
928
899
897
894
983
992
972
975
1370
1410
1360
1370
1470
1410
1410
1400
1840
1830
1800
1800
1990
1920
1910
1900
2360
2300
2300
2290
2390
2410
2340
2350
3310
2930
2820
2740
Even if the size of the modal equation is the same, the results produced by the mixed formulation [6.163] are not identical to those produced by the displacement formulation [6.150]. This is not so astonishing, since both methods are approximate in nature and based on a distinct formulation of the fluid-structure interaction. As a typical difference, if the model is restricted to the fluid and structural modes of rank zero, modal equation [6.150] leads to the low frequency mode of the rigid vessel with the fluid added mass vibrating about its supports and, as a numerical artefact, a high frequency mode related to the mass of the system and the stiffness coefficient used as a penalty factor to link the structure and the fluid together, as explained just above. According to the mixed model [6.163] we find the uniform pressure mode at zero frequency related to static displacement of the structure and the mode of the rigid vessel without the fluid added mass vibrating about its supports. Of course, this feature is a direct consequence of the mathematical formalism used in the coupled problem [6.159], which admits in particular the solution of the static version of [6.159], as already pointed out in subsection 6.2.1.3. Here the static problem reads as:
Vibroacoustic coupling
− Es Ss
∂2 X s + K 0 X sδ ( x ) + S f p ( δ ( x ) − δ ( x − L ) ) = 0 ∂x 2
529
[6.168]
∂2 p =0 ∂x 2
The modal solution is: ω =0
p = p0
;
;
Xs = −
p0 S f x
[6.169]
Es Ss
This particular solution being excepted, a close agreement can be achieved between the predictions of both models as soon as the discretization includes a sufficiently large number of modes, while comparison is restricted to modes of sufficiently low rank, as illustrated in Table 6.2. The results presented in Figures 6.28 and 6.31, are relevant to exactly the same mechanical system. However, the plots differ in nature as far as the fluid is concerned since Figure 6.28 show the displacement field in the fluid, whereas Figure 6.31 shows the pressure field. Finally, the problem can also be formulated as a symmetrical coupled system by using the pressure and potential displacement to describe the fluid. Again it is found appropriate to use dimensionless variables to formulate the problem. Starting from the system [6.140] and [6.141], the following differential equations are obtained: e ∂ 2X s ∂ 2X s ∂ 2φ F ( ) L + + − − = γ χ δ ξ δ ξ 1 ( ) ( ) ( ) f ∂ξ 2 ∂τ 2 ∂τ 2 Es S s
χ
∂ 2X s L2 S ( e ) ∂ 4φ ∂ 2ϖ − χ 2 + χ (δ (ξ ) − δ (ξ − 1) ) = 2 2 ∂τ ∂ξ ∂τ ∂τ 2 Es S s
ϖ−
[6.170]
∂ 2φ =0 ∂τ 2
where the following dimensionless quantities are used: Xs =
Xs L
⎛c ⎞ χ =⎜ e⎟ ⎝ cs ⎠
; φ= 2
Π ρ f L2
K ; γ0 = 0 Ks
; ϖ=
p ρ f ce2
; γf =
Kf Ks
; ξ= =
ρ f ce2 S f ρ s cs2 S s
x L =
; τ= Ef Sf
ce t ; L
[6.171]
Es S s
The modal problem is expressed in terms of the dimensionless frequency or wave number κ = ω L / ce as the symmetrical system:
530
−
Fluid-structure interaction
∂ 2X s + γ 0X sδ (ξ ) + κ 2 χ X s − γ f κ 2 (δ (ξ ) − δ (ξ − 1) ) φ = 0 ∂ξ 2
−γ f κ 2
∂ 2φ + γ f κ 2ϖ − γ f κ 2 (δ (ξ ) − δ (ξ − 1) ) X s = 0 ∂ξ 2
[6.172]
γ f κ 2φ + γ f ϖ = 0
The fields are expanded in the following modal series: N
N
N
n =0
n =0
n =0
X s = ∑ an cos ( nπξ ) ; φ = ∑ bn cos ( nπξ ) ; ϖ = ∑ cn cos ( nπξ )
Modal projection of [6.172] produces the
( 3N + 1) × ( 3N + 1)
[6.173]
matrix equation,
written in condensed form as: ⎡ [ Ass ] ⎡⎣ Asφ ⎤⎦ ⎢ ⎢ ⎡ Asφ ⎤ ⎡ Aφφ ⎤ ⎢⎣ ⎦ ⎣ ⎦ ⎢ [ 0] ⎡⎣ Aφ p ⎤⎦ ⎣
[ 0]
⎤ ⎥ ⎡[ a ]⎤ ⎡[ 0]⎤ ⎡⎣ Aφ p ⎤⎦ ⎥ ⎢⎢[b]⎥⎥ = ⎢⎢[ 0]⎥⎥ ⎥ ⎡⎣ App ⎤⎦ ⎥⎦ ⎢⎣ [ c ] ⎥⎦ ⎢⎣[ 0]⎥⎦
[6.174]
where the following matrices are defined as:
[ Ass ] = [ K ss ] − κ [ M ss ] 2
⎧⎪ K ss ( n, n ) = (γ 0 + n 2π 2 ) λn ; M ss ( n, n ) = χλn ; ⎨ K ss ( m, n ) = M ss ( m, n ) = 0 if m ≠ n ⎪⎩ N
(
⎡⎣ Asφ ⎤⎦ = −κ 2 ⎡⎣ M sφ ⎤⎦
; M sφ ( m, n ) = γ f ∑ 1 − ( −1)
⎡⎣ Aφφ ⎤⎦ = +κ 2 ⎡⎣ M φφ ⎤⎦
2 2 ⎪⎧ M ( n, n ) = π n γ f λn ; ⎨ φφ ⎪⎩ M φφ ( m, n ) = 0 if m ≠ n
2
⎡⎣ Aφ p ⎤⎦ = −κ ⎡⎣ M φ p ⎤⎦
n =0
⎧⎪ M φ p ( n, n ) = γ f λn ; ⎨ ⎪⎩ M φ p ( m, n ) = 0 if m ≠ n
m+n
) [6.175]
⎧⎪ K pp ( n, n ) = γ f λn ; ⎨ ⎪⎩ K pp ( m, n ) = 0 if m ≠ n ⎧ 1 if n = 0 with λn = ⎨ ⎩0.5 if n ≥ 1 ⎡⎣ App ⎤⎦ = ⎡⎣ K pp ⎤⎦
The major features of this model are similar to those of the mixed displacement pressure model, as indicated by comparing the natural frequencies reported in Tables 6.2 and 6.3, which are approximated values with three significant figures. It can be verified that the values given by the ( X s ; Π ; p ) formalism are in better
agreement with those given by the ( X s ; p ) than with those given by the ( X s ; X f
)
Vibroacoustic coupling
531
formalism, though relative discrepancy between the three methods is less than 1% as soon as the number of structural and fluid decoupled modes used is larger than a few tens. However as a specific feature of the ( X s ; Π ; p ) formalism, the matrix equation has N f +1 solutions at zero frequency (here N f = N ), which correspond to a mode shape where only the Π components differ from zero and are constant. These are the N f solutions already mentioned in subsection 6.2.1.4, while the last one arises because of the boundary conditions of the present problem. Table 6.3. Numerical approximations of the natural frequencies ( X s ; Π ; p ) versus ( X s ; p ) variables (accuracy of about 0. 5%)
Mixed formulation
Mixed formulation ( X s ; p)
Mixed formulation ( X s ; Π ; p)
Mixed formulation ( X s ; p)
N s = N f = 10
N s = N f = 10
N s = N f = 50
N s = N f = 50
Frequency (Hz)
Frequency (Hz)
Frequency (Hz)
Frequency (Hz)
(Xs ; Π
; p)
0
0
—
0
97.4
97
96.5
96.5
430
430
424
424
512
511
509
508
899
899
894
894
994
992
977
975
1410
1410
1380
1370
1410
1410
1400
1400
1830
1830
1800
1800
1920
1920
1910
1900
2310
2300
2290
2290
2410
2410
2350
2350
2930
2930
2750
2740
6.4.1.3 Vibroacoustic modes of an inflated toroidal shell We consider here the toroidal ring of Figure 6.32 which can stand typically for the inner tube of a tyre. It is recalled that the in-plane flexure modes of the equivalent ring in vacuum are governed with a good degree of accuracy by the following radial equation (cf. [AXI 05], Chapter 8):
532
Fluid-structure interaction
⎞ Es I ⎛ ∂ 4U s ∂ 2U s + + U s ⎟ − ω 2 ρ s S sU s = 0 2 ⎜ 4 2 4 R ⎝ ∂θ ∂θ ⎠
[6.176]
where U s is the radial displacement of the beam-like model of the structure and Es I its in-plane flexural rigidity.
Figure 6.32. Pressurized toroidal tube
In the context of the present exercise, this equation must be modified into two distinct directions. First, the coupling of the tube with the inner fluid is accounted for in accordance with equations [6.138] and [6.139]: pS f ⎞ ∂ 2U s Es I ⎛ ∂ 4U s +2 + U s ⎟ − ω 2 ( ρ s Ss + ρ f S f )U s − =0 4 ⎜ 4 2 ∂θ R ⎝ ∂θ R ⎠ U 1 ∂ 2 p ω2 + p + ω2ρ f s = 0 R 2 ∂ θ 2 ce2 R
[6.177]
Furthermore, if the tyre is pressurized at P0 above the external pressure, it vibrates about a pre-stressed state and the corresponding stiffness terms are to be included into the model. It is recalled that according to the analysis made in [AXI 05], Chapter 7, P0 induces the elastic tensile stresses: N θθ
P0 a 1 − ν s2
; Nϕϕ
P0 a 2 (1 − ν s2 )
[6.178]
Vibroacoustic coupling
533
Therefore, the equivalent ring is tensioned by the tangential and uniform tensile force: T0
2π a 2 P0 1 − ν s2
[6.179]
where ν s designates the Poisson ratio. This corresponds essentially to twice the resulting pressure force which would be exerted on a meridian cross-section of the torus occupied by the fluid. As shorthand to derive the effect of this preload on the bending equation, the Newtonian approach (see [AXI 05], Chapter 3) is used here instead of the variational method. The ring equation [6.177] is rewritten in the time domain as a force balance without specifying the material law which controls the stresses: 1 ∂Q M z ρ s SsUs − − 2 = Fr( e ) R ∂θ R
[6.180]
Q stands for the in-plane shear force and M z for the bending moment about the Oz axis, perpendicular to the equatorial plane of the torus. It is also recalled that the small flexure angle is related to the radial U s and tangential displacements Vs of the ring by the relation:
ψz =
1 ∂U s Vs − R ∂θ R
[6.181]
This angle induces a pre-load component in the radial direction which must be added to the elastic shear force: Q=−
Es I ∂ 3U s T0 ⎛ ∂U s ⎞ + ⎜ − Vs ⎟ R 3 ∂θ 3 R ⎝ ∂θ ⎠
[6.182]
Substituting the shear force back into [6.182], the following equation is obtained: ⎞ T0 ⎛ ∂ 2U s ∂ Vs Es I ⎛ ∂ 4U s ∂ 2U s + + U − 2 ⎜ s⎟− 2 ⎜ 2 R4 ⎝ ∂ θ 4 ∂θ2 ∂θ ⎠ R ⎝ ∂θ
⎞ ⎟ + ρ s S sU s = 0 ⎠
[6.183]
Finally, the tangential displacement is eliminated from [6.183] by assuming that flexure is not coupled to traction, which turns out to be the case to a high degree of accuracy, as shown in [AXI 05], Chapter 8. Accordingly the global equatorial strain is zero: ηθθ =
1 ∂ Vs U s + =0 R ∂θ R
Whence the final form of the coupled equations used in the present study:
[6.184]
534
Fluid-structure interaction
pS f ⎞ T0 ⎛ ∂ 2U s ⎞ ∂ 2U s Es I ⎛ ∂ 4U s + + + U s ⎟ − ω 2 ( ρ s Ss + ρ f S f )U s − =0 U 2 ⎜ ⎟− 2 ⎜ s 4 4 2 2 ∂θ R ⎝ ∂θ R ⎠ R ⎝ ∂θ ⎠ U 1 ∂ 2 p ω2 + 2 p + ω2ρ f s = 0 2 2 R ∂θ ce R
[6.185] The system [6.185] is discretized by expanding U s and p as Fourier series of the even type, where the breathing mode n = 0, is discarded: NS
U s = ∑ an cos nθ
;
n =1
Nf
p = ∑ bn cos nθ
[6.186]
n =1
By virtue of the orthogonality properties of the mode shapes, there is no coupling between modes of distinct rank. The modal matrix equation for the coupled mode of rank n is: ⎡E I T0 ( n 2 − 1) 2 2 s − ω 2 me ⎢ 4 ( n − 1) + 2 R ⎢R ⎢ ρ ⎢ −ω 2 f ⎢⎣ R
⎤ ⎥ R ⎥ ⎡ an ⎤ = ⎡ 0 ⎤ 2⎥⎢ ⎥ ⎢ ⎥ 2 ⎛ n ⎞ ⎛ ω ⎞ ⎥ ⎣ bn ⎦ ⎣0⎦ − ⎜ ⎟ ⎜ ⎟ ⎥ ⎝ R ⎠ ⎝ ce ⎠ ⎦ −
Sf
[6.187]
where me = ρ s S s + ρ f S f = ms + m f is the equivalent mass per unit length of the tube and the contained fluid. A particular case of interest is the mode n = 1 which in the absence of fluid corresponds to the rigid mode of the torus translated in the equatorial plane along the direction θ = 0. This mode is at zero frequency if the torus is not provided with elastic supports. However, due to the vibroacoustic coupling, a mode n = 1 arises whose frequency is not zero. The characteristic polynomial related to the modal equation [6.187] reads as: 2 ⎛ ⎛ ω ⎞2 1 ⎞ ω 2 ( me + m f ) ⎛ω ⎞ 2 ω me ⎜ ⎜ ⎟ − 2 ⎟ − + ω me ⎜ ⎟ = 0 ⎜ ⎝ ce ⎠ R ⎟ R2 ⎝ ce ⎠ ⎝ ⎠ 2
[6.188]
The solution ω = 0 corresponds to the eigenvector [1 0] , which means a static T
rigid displacement and no fluctuating pressure, which is natural since there is no fluid acceleration. The second solution is: ω1 =
mf ce 1+ R ms + m f
The corresponding eigenvector is:
[6.189]
Vibroacoustic coupling
⎡ ⎛ m 2 ⎢1 − ρ f ce ⎜⎜ 2 + s mf ⎢⎣ ⎝
⎞⎤ ⎟⎟ ⎥ ⎠ ⎥⎦
535
T
[6.190]
The shape [6.190] attests thus to the vibroacoustic nature of the mode. The restoring force leading to the stiffness coefficient is provided by the pressure force which is in phase opposition from one half of the torus to the other. This is a remarkable result in that it combines the structural and fluid properties in such a way the natural frequency is proportional to the speed of sound in the fluid and the square root of a mass ratio where the fluid term appears both in the numerator and the denominator.
Figure 6.33. Vibroacoustic coupling
536
Fluid-structure interaction
Besides this particular mode, vibroacoustic coupling is expected to be significant near the crossing of the natural frequency plots of the uncoupled system. The upper frequency plot of Figure 6.33 refers to a torus of stainless steel R = 1 m, a = 10 cm, h = 2 mm, filled with water ρ f = 1000 kg/m 3 ; c f =1500 m/s . Accordingly, m f 31 kg , ms 10 kg . The plot displays the vibroacoustic modal frequencies
versus the rank n. The full and the dashed lines refer to the upper and lower branches of the coupled computation respectively. Coupling is most significant for the vibroacoustic modes n = 1 and n = 6 . The uncoupled n = 1 acoustic mode at 166 Hz gives rise to the n = 1 vibroacoustic mode at 220 Hz. On the other hand, by inspecting Table 6.4 it can be verified that the closest pair of uncoupled natural frequencies corresponds to the sixth acoustical mode at 995 Hz and to the sixth structural mode at 1011 Hz. If a less rigid and dense material is used instead of stainless steel, for instance Perpex ( Es = 5.6109 Pa and ρ s = 1200 kg/ m 3 ), coupling is even more important, see lower plot of Figure 6.33. It is also noted that the equivalent speed of sound of water is considerably lowered due to the relatively large flexibility of the breathing mode (n = 0) of the tube. Table 6.4. Natural frequencies of the torus
Coupled modes: lower branch frequency (Hz)
Uncoupled modes: upper branch frequency (Hz)
Coupled modes: upper branch frequency (Hz)
0 (S)
0 (S)
166 (A)
220
86.6 (S)
79.0
332 (A)
363
231 (S)
220
498 (A)
523
433 (S)
417
663 (A)
689
693 (S)
665
829 (A)
864
995 (A)
933
1011 (S)
1080
1160 (A)
1140
1390 (S)
1410
1330 (A)
1320
1820 (S)
1830
1490 (A)
1499
2310 (S)
2320
1660 (A)
1660
2860 (S)
2860
Uncoupled modes: lower branch frequency (Hz)
To investigate the relative importance of pressurization and vibroacoustic coupling, we consider a torus R = 60 cm, a = 1 cm, h = 1 mm made of soft rubber ( Es = 5.106 Pa and ρ s = 950 kg/ m 3 ) pressurized at 1 bar by using first air and then an air-water mixture. Hoop stress is σ 0 = 1 MPa , dilatation is 2.4 cm in the
Vibroacoustic coupling
537
equatorial plane and 1.7 mm in a meridian plane. It can be easily anticipated that the torus inflated with dry air is typical of a case of very poor vibroacoustic coupling for the two following reasons. First, the mass ratio is very small μ f = 0.014 . Then, the speed of sound in the enclosed air is much larger than the structural wave speed due to tension cT0 = T0 / me (235 m/s instead of 31 m/s). In order to obtain cT0 ce , it would be necessary to increase the static pressure up to about 100 bar, which is very unrealistic.
Figure 6.34a. Pressurized torus filled with dry air. Upper plot: upper and lower frequency branches with and without pressurization. Lower plot: dimensionless pressure to displacement ratios corresponding to the upper and lower frequency branches
538
Fluid-structure interaction
Figure 6.34b. Pressurized torus filled with air-water mixture. Upper plot: upper and lower frequency branches with and without pressurization. Lower plot: dimensionless pressure to displacement ratios corresponding to the upper and lower frequency branches
Computed modal features are summarized in Figure 6.34a. As in Figure 6.33, the upper plot displays the vibroacoustic modal frequencies versus the rank n. The full and the dashed lines refer to the upper and lower branches of the coupled computation respectively. Importance of pressurization can be checked by comparing the two low frequency branches for P0 = 1 bar and P0 = 0 bar . The non pressurized case corresponds to the dotted dashed line marked by circles. It is substantially lower than the dashed line which corresponds to P0 = 1 bar . On the other hand, the upper branch is much higher than the low branches indicating clearly
Vibroacoustic coupling
539
that vibroacoustic coupling can be safely neglected even in the pressurized case. The fact that the upper branch corresponds to acoustic modes, whereas the low branches correspond to structural modes can be checked further by inspecting the lower plot of Figure 6.34a, which displays the ratio of modal pressure to modal displacement. Radial displacement is scaled by the radius of the torus and pressure is scaled by the pressure component of the mode shape [6.190]. Here again, the dashed line corresponds to the low frequency branch and is found to induce practically no pressure. The full line which refers to the upper frequency branch is found to induce negligible displacement except at n = 1. As a further exercise of academic interest, to increase significantly the fluid to solid mass ratio and to achieve the condition cT0 ce the pressurized torus is filled with an air-water homogeneous mixture with a void ratio of about 0.5 such that ρ f 500 kgm -3 and c f 35 ms -1 (cf. Chapter 4, subsection 4.4.3). The natural frequencies of the coupled modes are essentially the same as those of the uncoupled modes; however, coupling is significantly more important than in the former case, whatever n may be, as it can be verified by inspecting the results displayed in Figure 6.34b. 6.4.1.4 Thermal expansion lyre filled with incompressible fluid
Figure 6.35. In-plane rocking mode shape of the thermal expansion lyre
Industrial pipe works conveying hot fluids are often provided by curved parts like U shaped tubes or lyres to accommodate thermal expansion stresses by increasing significantly their flexibility. Such a flexible part shaped as a lyre is sketched in Figure 6.35. It depicts both the non deformed configuration of the pipe and the simplified shape of the so-called in-plane rocking mode. The straight parts (OA) and (DE) are supposed to stay practically at rest, whereas the upper part (BC)
540
Fluid-structure interaction
is translated in the longitudinal direction. Finally, the straight parts (AB) and (DC) rotate rigidly about (A) and (D), respectively. According to such an idealized shape, the elastic strains are concentrated at the 90° bends of the lyre. The high flexibility of thin walled bended tubes provides the device with the desired property to act as a thermal expansion joint, justifying also the present model. To simplify further the analytical calculation, we assume that the frequency of the rocking mode is so low that ω L / ce << 1 , where L is the total length of the pipe. As a consequence, the vibroacoustic coupling mechanism reduces here to inertial coupling and the problem actually solved is to determine the appropriate fluid added mass coefficient associated to the rocking mode of the pipe, based on the coupled equations [6.138] and [6.139].
Figure 6.36. Coupling forces exerted on the upper elbows
A first contribution to the added mass is due to the inertial transverse effect acting on the segments (AB) and (DC), which is: H
M at =
⌠ ⎮ 2 ρ f S f ⎮⎮ ⎮ ⌡0
2
2 ⎛ z ⎞ ⎜ ⎟ dz = M f 3 ⎝H⎠
[6.191]
where M f = ρ f S f H . The other contribution to the added mass coefficient is related to the longitudinal coupling between the fluid column and the pipe which occurs at the curved ends of the segment (BC). By virtue of the same reasoning as in subsection 6.3.2.3, the resulting force exerted by the fluid on the elbows (B) and (C) is found to have a longitudinal component: Fx = [ pC − pB ] S f i [6.192]
Vibroacoustic coupling
541
The transverse component is Fz = ( pB + pC ) S f k , see Figure 6.36. Of course,
similar (but different) forces are also exerted on the lower elbows (A) and (D). However, since the fluid-structure coupling is related to the generalized force exerted by the fluid on the pipe vibrating according to the rocking mode, the only force component to be considered here is Fx = [ pC − pB ] S f i as the others vanish once projected onto the rocking mode shape. Conversely, the motion of the elbows (B) and (C) induces a volume velocity source. In agreement with [6.136], they are expressed as: QB = +iω S f X 0
; QC = −iω S f X 0
[6.193]
where X 0 is the displacement at B and C.
Figure 6.37. Equivalent acoustical circuit (plane wave model)
The problem is thus to determine the pressure field induced by the sources [6.193] in the pipe work. The solution is unique provided suitable terminal impedances are prescribed at the ends of the pipe. The task to calculate it effectively can be alleviated by using the acoustical transfer matrix technique already introduced in Chapter 4. The equivalent acoustical circuit is depicted in Figure 6.37. Taking, as an example, the case where the pipe is connecting two large enclosures, the terminal impedances can be idealized as pressure nodes, at least as a first approximation. Using the low frequency asymptotic form [4.133] of the acoustic transfer matrix for a uniform tube element, we obtain: 1 0⎤ ⎡ ⎡ qB ⎤ ⎢ ⎥ ⎡ qo ⎤ ⎢ p ⎥ = ⎢ − iωρ f ( L1 + H ) 1 ⎥ ⎢ 0 ⎥ ⎣ B⎦ ⎢ ⎥⎣ ⎦ Sf ⎣ ⎦
[6.194]
1 ⎡ ⎡ qC ⎤ ⎢ ⎢ p ⎥ = ⎢ − iωρ f L2 ⎣ C⎦ ⎢ Sf ⎣
[6.195]
0⎤ ⎥ ⎡ qB + QB ⎤ 1 ⎥ ⎣⎢ pB ⎦⎥ ⎥ ⎦
542
Fluid-structure interaction
1 0⎤ ⎡ ⎡ qE ⎤ ⎢ ⎥ ⎡ qB + QB + QC ⎤ ⎥ ⎢ 0 ⎥ = ⎢ − iωρ f ( L1 + H ) 1 ⎥ ⎢ pC ⎣ ⎦ ⎢ ⎦ ⎥⎣ S f ⎣ ⎦
[6.196]
Whence: pB = −
iωρ f Sf
( L1 + H ) qO ;
pC =
ωρ f iS f
( L2 + L1 + H ) qO + ω 2 ρ f L2 X 0 ;
qO = −
iω S f L2 X 0
L2 + 2 ( L1 + H )
leading to: pB = −
ω 2 ρ f L2 ( L1 + H ) L2 + 2 ( L1 + H )
X 0 ; pC = ω 2 ρ f L2
( L2 + H ) X 0 L2 + 2 ( L1 + H )
[6.197]
The generalized force exerted by the fluid on the pipe vibrating according to the rocking mode is: Fx X 0 = −ω 2 M va X 02
where M va = m f
[6.198] 2 ( L1 + H )
L2 + 2 ( L1 + H )
and m f = ρ f L2 S f .
Figure 6.38. Pressure and volume velocity associated with the rocking motion of the lyre profiles along the circuit provided with two pressure nodes at the ends
Vibroacoustic coupling
543
If L2 is much smaller than L1 + H , M va tends to m f = L2 ρ f S f and thus becomes negligible, which is the expected result since the resulting coupling force is proportional to the pressure difference between B and C. On the other hand, if L2 is
much larger than L1 + H , M va tends to the constant value 2 ρ f S f ( L1 + H ) , which means that the volume velocity induced at the elbows excites an oscillation of the incompressible water column in the shortest part of the circuit only, that is between (O) and (B) on one side and between (E) and (C) on the other. The fluid does not move in the segment (BC) which acts as a stopping barrier because of its large inertia. Whatever the value of L2 may be, M va differs from the physical mass of the fluid contained in the segment (BC). The partial results just established can be finalized as the resulting added mass coefficient: M a = M at + M va =
2 ( L1 + H ) m f 2 Mf + L2 + 2 ( L1 + H ) 3
[6.199]
The corresponding profiles along the tube of the fluctuating pressure and volume velocity are displayed in Figure 6.38. Of course, if the pipe is provided with other terminal impedances than the pressure nodes assumed in the above calculation, the pressure and volume velocity profiles are modified and so is the coupling force. For instance, if the large enclosures are modelled as elastic impedances in agreement with the low frequency matrix [4.134], it is found that the coupling force comprises both an inertial and an elastic component. It is left to the reader as an exercise to establish that if the volume VE of the enclosure is sufficiently large, the inertia force is the same as the value [6.199] while the added stiffness coefficient is: K va = 2
ρ f c 2f S f
[6.200]
VE
On the other hand, if the pipe is closed at both ends, the force is again purely inertial in nature and the corresponding added mass coefficient is found to be: M a = M at + M va =
2 M f + mf 3
[6.201]
The result [6.201] could be anticipated since if the ends of the pipe are closed, the incompressible fluid is prevented moving in the segments (OA) and (DE). 6.4.1.5 Thermal expansion lyre filled with compressible fluid To extend the preceding problem to the compressible case is straightforward, since the calculation process goes along the same lines as in the incompressible case. The task of producing the solution is however tedious because the coefficients of the acoustical matrices are more complicated than those of the incompressible case. Furthermore, the actual calculation soon becomes impracticable as the complexity of the pipe geometry is increased. Therefore, it is preferable to devise numerical techniques for solving this kind of problem. The method presented here extends to
544
Fluid-structure interaction
one dimensional vibroacoustic problems the transfer matrix method, described earlier in the context of plane wave acoustics. Turning back to the example of the thermal expansion lyre, the pipe work is discretized into three segments (OB), (BC) and (CE), described by the acoustic transfer matrices denoted [ a ] , [ b] and [ c ] respectively. They are assembled by using the procedure presented in Chapter 4. To model the structural part of the system, namely here the rocking mode of the pipe, the acoustic matrix equation is provided with an additional row which describes the dynamical equation of the modal oscillator corresponding to the rocking mode of the lyre, expressed in terms of the displacement variable X 0 as:
(K
s
− ω 2 M e ) X 0 = F (ω )
; M e = M s + M at
; M at =
2 ρf Sf H 3
[6.202]
The fluid and the solid parts of the system are then coupled together by formulating the volume velocity sources related to the motion of the structure and the pressure forces exerted by the fluid on the structures as internal excitations. In this way a vibroacoustic matrix equation is obtained. For the present problem, assuming the top of the lyre is excited by a harmonic longitudinal force (direction i ) of unit amplitude, the matrix equation reads as: ⎡ Ze −1 0 0 0 0 ⎢ 0 ⎢ a11 a12 −1 0 0 ⎢ a21 a22 0 −1 0 0 ⎢ 0 1 0 −1 0 ⎢0 ⎢0 0 0 1 0 −1 ⎢ 0 0 0 b11 b12 ⎢0 ⎢0 0 0 0 b21 b22 ⎢ 0 0 0 0 0 ⎢0 ⎢ 0 0 0 0 0 0 ⎢ ⎢0 0 0 0 0 0 ⎢ 0 0 0 0 0 0 ⎢ ⎢0 0 0 0 0 0 ⎢ 0 0 0 0 +S f ⎢⎣ 0
0
0
0
0
0
0
0
0
0 0
0 0
0 0
0 0
0
0
0
0
−1 0 0 −1
0 0
0 0
1
0
−1
0
0
1
0
−1
0 0
0 0
c11 c21
c12 c22
0
0
0
0
0
0
0
−S f
0
⎤ ⎡ q1 ⎤ ⎡ 0⎤ ⎥⎢ ⎥ ⎢ ⎥ 0 0 ⎥ ⎢ p1 ⎥ ⎢ 0⎥ ⎥ ⎢ q ⎥ ⎢ 0⎥ 0 0 0 ⎥⎢ 2 ⎥ ⎢ ⎥ iωρ f S f 0 0 ⎥ ⎢ p2 ⎥ ⎢ 0⎥ ⎥ ⎢ q ⎥ ⎢ 0⎥ 0 0 0 ⎥⎢ 3 ⎥ ⎢ ⎥ 0 0 0 ⎥ ⎢ p3 ⎥ ⎢ 0⎥ ⎥ ⎢ q ⎥ = ⎢ 0⎥ 0 0 0 ⎥⎢ 4 ⎥ ⎢ ⎥ −iωρ f S f 0 0 ⎥ ⎢ p4 ⎥ ⎢ 0⎥ ⎥⎢ ⎥ ⎢ ⎥ 0 0 0 ⎥ ⎢ q5 ⎥ ⎢ 0⎥ 0 ⎥ −1 0 ⎢ p5 ⎥ ⎢ 0⎥ ⎥⎢ ⎥ ⎢ ⎥ 0 0 −1 ⎥ ⎢ q6 ⎥ ⎢ 0⎥ ⎥ ⎢ p ⎥ ⎢ 0⎥ 0 Zs −1 ⎥⎢ 6 ⎥ ⎢ ⎥ 0 0 ( K s + iω Cs − ω 2 M e ) ⎥ ⎣ X 0 ⎦ ⎣1⎦ ⎦ 0
0
0
[6.203] The vibroacoustic resonances are then analysed by inspecting the transfer functions produced by solving the forced equation [6.203]. The computational method is thus mimicking closely the techniques of experimental modal analysis. As an illustration, Figure 6.39 displays a few response spectra of the displacement and pressure at the elbows (B) or (C) of a lyre which contains liquid water and is terminated by two dissipative pressure nodes. The geometry of the stainless steel
Vibroacoustic coupling
545
tube is as follows: Total length L = 100 m, L1 = H = 22.5 m, L2 = 10 m , thickness h = 2 mm and radius R = 20 cm.
Figure 6.39. Vibroacoustic response displacement and pressure spectra at the elbows (B) and (C)
The top of the lyre is excited by a harmonic force of magnitude 1 kN, and frequency swept between 0.1 and 60 Hz. The natural frequency of the rocking mode is supposed to be 26.8 Hz in vacuum and the damping ratio is ς v = 0.06 in such a way that in incompressible water the natural frequency would be 10 Hz and the damping ratio would be ς f = 0.02 . From the upper left to the lower right plot, the spectra shown in Figure 6.39 relate to the following computational cases: 1. Rocking mode at 10 Hz in water (about 26.8 Hz in vacuum) and incompressible fluid (first acoustic resonance artificially set to 500 Hz). As expected, in the frequency range explored the response spectrum presents a single peak at 10 Hz. The r.m.s values of the displacement and pressure are σ XX 2.5 mm and σ pp 0.48 bar . 2. Rocking mode at 10 Hz in water and compressible fluid (first acoustic resonance 5 Hz). In the vicinity of the natural frequency of the rocking mode, which is equal to that of the first antisymmetrical acoustic mode,
546
Fluid-structure interaction
vibroacoustic coupling induces a splitting of the structural resonance into two distinct response peaks which are much sharper and of higher magnitude than the corresponding peak observed in incompressible fluid. At higher frequencies, several peaks of much lower amplitude are also observed which can be recognized as antisymetrical acoustic modes slightly coupled to the rocking mode. The r.m.s. values of the displacement and pressure are significantly larger than in the incompressible case: σ XX 5.8 mm and σ pp 1.3 bar .
Figure 6.40. Vibroacoustic response: displacement and pressure spectra at the elbows (B) and (C)
Figure 6.40 displays the corresponding information for the following geometry: Total length L = 100 m, L1 = H = 1 m, L2 = 96 m , thickness h = 2 mm and radius R = 20 cm. Again, the natural frequency of the rocking mode is set to 26.8 Hz in vacuum. In incompressible water the natural frequency would be 23.5 Hz and the damping ratio would be ς f = 0.02 . In the incompressible case, the r.m.s. value of the displacement is significantly less than in the first geometry, σ XX ≅ 0.87 mm instead of 2.5 mm , whereas the r.m.s. value of the fluctuating pressure is much less changed: σ pp ≅ 0.37 bar instead of 0.48 bar. Finally, if fluid compressibility is accounted for, sharp acoustic peaks arise below and above the broad peak centred at
Vibroacoustic coupling
547
about 23.5 Hz. The r.m.s. value of the displacement remains essentially the same as in the incompressible case, though the r.m.s. value of the fluctuating pressure is changed. Such results indicate that vibroacoustic coupling is much less efficient than in the first geometry.
Figure 6.41. Displacement spectra at the elbows induced by acoustical sources concentrated at the inlet of the lyre. Pressure source (upper plot) and a mass flow rate source (lower plot)
548
Fluid-structure interaction
To conclude this exercise, it is also interesting to analyse the response of the lyre to an external acoustical source, for instance a pressure source concentrated at the inlet provided with a pressure node, or a volume velocity source also concentrated at the inlet and provided with a pressure antinode. To simplify the interpretation of the results, source amplitude is assumed to be constant in the range of frequencies of interest. The spectrum of the response to a pressure source presents two major peaks corresponding to the vibroacoustic coupling between the rocking mode and the second acoustic mode, see Figure 6.41. Both of them are at about 10 Hz, and the two separated peaks arise as a consequence of coupling. The other peaks are interpreted as the forced response of the lyre to the acoustic resonances. It is noted that the structure is excited by the antisymmetrical modes only, which is not surprising as the resultant of the forces exerted on the two elbows vanish if pressure is the same at (B) and at (C) as it is the case for symmetrical modes. The response spectrum in the right-hand side is induced by a mass flow-rate source of about 8 kg/s. It can correspond to a piston vibrating at the inlet with amplitude of 1 mm at a frequency of about 10 Hz. The terminal impedances are assumed to be α in = π / 2 + nπ and α out = nπ . Hence the first acoustic resonance is at 2.5 Hz instead of 5 Hz and the symmetry about the central part of the lyre disappears, so far as the acoustic part of the problem is concerned. As a consequence, all the successive acoustical harmonics are present in the response spectrum, whose largest peak corresponds to the rocking mode of the lyre at 10 Hz. 6.4.2
Simplified model of a drum using modal expansions
Musical drums, whose sound we experience everyday, are an interesting and relevant application of the previously introduced concepts. The structural component of these systems – a thin membrane made from calf skin or mylar (polyethylene terephthalate), a light and very stable material with respect to changes in temperature and humidity – is relatively light, when compared to the mass of the interacting air. Therefore, one should expect the dynamical behaviour of the system to be significantly affected by vibroacoustic coupling. Orchestral kettledrums, or timpani, are drums which convey a strong sense of pitch due to adequate tuning of a few low order modes of the system, in order to obtain a near-harmonic sequence. This is achieved by adjusting the tension of the membrane, before concerts, but also through a clever design of the kettle. Indeed, in vacuum, the modal frequencies of the circular membranes used in timpani do not display near-integer ratios, as they are related to the roots of Bessel functions (see Chapter 4, subsection 4.2.1.10). This can however be corrected (for the first three modes, or so) by a proper design of the air cavity, so that the coupled vibroacoustic modes display the intended frequency ratios. Here, for illustration, we will model a basic drum somewhat remote from the original problem, consisting on a rectangular membrane tensioned over a rectangular enclosure with rigid walls (the drum “kettle”), which allows for a less involved analytical formulation. As shown in Figure 6.42, The dimensions of the membrane are Lx and Ly , the height of the resonating cavity being Lz . First we will address
Vibroacoustic coupling
549
the membrane/cavity interaction, assumed conservative, deferring a more realistic solution which also must include the interaction of the membrane with the external fluid and radiation phenomena.
T
x
T
mS
T
T
y
ρ f , cf z
Lz
Ly
Lx Figure 6.42. Tensioned rectangular membrane over a resonating cavity
The problem is solved by using a mixed modal expansion, as explained in subsections 6.2.1.1 and 6.2.1.3. From [6.10] the coupled equations of the membrane and enclosure are written as: ⎛ ∂2Z ∂2Z ⎞ T ⎜ 2s + 2s ⎟ + msω 2 Z s = p ( x, y ,0; ω ) ∂y ⎠ ⎝ ∂x ∂2 p ∂2 p ∂2 p ⎛ ω + + +⎜ ∂x 2 ∂y 2 ∂z 2 ⎜⎝ c f
2
⎞ 2 ⎟⎟ p = ω ρ f Z s ( x, y; ω ) δ ( z ) ⎠
[6.204]
with pressure antinodes assumed at the enclosure walls and bottom. The membrane equation can also be written as: 2
∂2Zs ∂2Zs ⎛ ω ⎞ 1 + 2 + ⎜ ⎟ Z s = p ( x, y ,0; ω ) ∂x 2 ∂y c T ⎝ s⎠
[6.205]
cs = T ms is the propagating velocity of the transverse waves, where T [N/m] is
the uniform tensile force and ms [Kg/m2] is the membrane mass density per unit
550
Fluid-structure interaction
area (see for instance [AXI 05], Chapter 6). In order to solve the coupled problem [6.204], we describe the membrane motion in terms of its transverse modes: 2 2 ⎧ ⎪ ω = π c ⎛ m ⎞ + ⎛⎜ n ⎞⎟ ⎟ ⎜ ⎟ s ⎜ ⎪ mn ⎝ Lx ⎠ ⎝ Ly ⎠ ⎨ ⎪ mπ x nπ y sin ⎪φmn ( x, y ) = sin Lx Ly ⎩
m, n = 1, 2,3,...
[6.206]
and, as discussed in subsection 6.2.1.3, the enclosure response is described in terms of the acoustical modes of a box with a rigid boundary condition assumed at the coupling interface: 2 2 2 ⎧ ⎪ ω = π c ⎛ p ⎞ + ⎜⎛ q ⎟⎞ + ⎛ r ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ f ⎪⎪ pqr ⎝ Lx ⎠ ⎝ Ly ⎠ ⎝ Lz ⎠ ⎨ ⎪ ( p) pπ x qπ y rπ z cos cos ⎪ϕ pqr ( x, y, z ) = cos Lx Ly Lz ⎪⎩
;
p, q, r = 0,1, 2,...
[6.207]
The physical responses of the membrane and of the acoustic volume are given respectively by the displacement and pressure fields: M
N
Z s = ∑∑ φmn ( x, y ) amn
;
m =1 n =1
Q
P
R
( p) p = ∑∑∑ ϕ pqr ( x, y , z ) bpqr
[6.208]
p =0 q =0 r =0
Then, replacing [6.208] in [6.204] and performing the modal projections, we obtain: ⎛ ⎡[ K mn ] [ 0 ] ⎤ ⎡[ M mn ] [ 0 ] ⎤ ⎞ ⎧⎪{amn } ⎫⎪ ⎪⎧ { Fmn } ⎪⎫ ⎜⎢ −ω2 ⎢ ⎥ ⎥⎟⎨ ⎬=⎨ ⎬ ⎜ ⎢ [ 0 ] ⎡⎣ K pqr ⎤⎦ ⎥ ⎡⎣ M pqr ⎤⎦ ⎥ ⎟ ⎪{bpqr }⎪ ⎪{ Fpqr }⎪ 0 [ ] ⎢ ⎩ ⎭ ⎩ ⎦ ⎣ ⎦⎠ ⎭ ⎝⎣
[6.209]
with the membrane modal coefficients given as: Lx L y
M mn = ms ∫
∫ [φ
0 0
( x, y )] dxdy = 2
mn
K mn = M mn ω
=
4
π T (m L + n L 2
2 mn
mS Lx Ly
2 2 y
2 2 x
4 Lx Ly
)
[6.210]
and the acoustical modal coefficients: 1 M pqr = 2 cf
Lx L y Lz
∫∫∫ 0 0 0
K pqr = M pqr ω
2 pqr
Lx Ly Lz 2 ( p) ⎡⎣ϕ pqr ( x, y, z ) ⎤⎦ dxdydz = k 2 2 cf =
π 2 ( p 2 L2y L2z + q 2 L2x L2z + r 2 L2x L2y ) 2k Lx Ly Lz
[6.211]
Vibroacoustic coupling
551
where 0 ≤ k ≤ 3 is the number of non-zero indexes p, q, r . On the other hand, the vibroacoustical coupling terms are computed as: Fmn =
Lx L y
P
∫∫
R
p =0 q =0 r =0
0 0
Fpqr = ω 2 ρ f
Q
pqr p( x, y ,0; ω ) φmn ( x, y ) dxdy = ∑∑∑ Cmn bpqr
Lx L y
∫ ∫ Z ( x, y; ω ) ϕ s
0 0
( p) pqr
M
N
[6.212]
mn ( x, y ,0) dxdy = ω ρ f ∑∑ C pqr amn 2
m =1 n =1
where the coupling coefficients are given by: pqr Cmn =
Lx L y
∫ ∫ϕ
( p) pqr
( x, y, 0) φmn ( x, y ) dxdy
0 0
C
mn pqr
[6.213]
Lx L y
=
∫ ∫φ
mn
( x, y ) ϕ
( p) pqr
( x, y, 0) dxdy
0 0
mn pqr T ⎤⎦ = ⎡⎣Cmn ⎤⎦ . After integration we obtain: therefore ⎡⎣C pqr
pqr mn Cmn = C pqr
⎧ ⎪ =⎨ ⎪ ⎩
Lx Ly π2
mn
1− (−1)m+ p − (−1)n+q + (−1)m+n+ p+q (m ≠ p , n ≠ q) (m2 − p2 )(n2 − q2 ) 0 (m = p and/or n = q)
[6.214]
Hence [6.209] and [6.212] lead to the modal equation of the vibroacoustic system: pqr ⎛ ⎡[ Kmn ] − ⎡Cmn ⎤⎦ ⎤ ⎡ [ M mn ] [ 0 ] ⎤ ⎞⎟ ⎪⎧{amn } ⎪⎫ ⎧⎪{ 0 }⎫⎪ ⎣ ⎜⎢ ⎥ −ω2 ⎢ ⎥ ⎨ ⎬=⎨ ⎬ mn ⎜ ⎢ [ 0 ] ⎡ K ⎤ ⎥ ⎢ ρ f ⎣⎡C pqr ⎦⎤ ⎣⎡ M pqr ⎦⎤ ⎦⎥ ⎟ ⎩⎪{bpqr }⎭⎪ ⎪⎩{ 0 }⎪⎭ pqr ⎦ ⎦ ⎣ ⎣ ⎣ ⎝ ⎠
[6.215]
This, as previously discussed, is a typical mixed formulation asymmetrical eigenproblem. We will now illustrate the results of the above formulation, on a drum with a square mylar membrane Lx = Ly = 0.5 m, for a range of values of the enclosure height Lz = 0.1 ~ 1.0 m. The numerical parameters used for the membrane are T = 4000 N/m and ms = 0.25 Kg/m2 (or cs = 126.5 m/s), all of these representative of current timpani, while for air we have ρ f = 1.21 Kg/m3 and c f = 343 m/s. Figure 6.43 displays the change of the modal frequencies of the membrane, in vacuum, as well as those of the acoustic modes of the enclosure (rigidly closed at the membrane interface) as a function of the box height Lz . Of course, only the longitudinal and mixed acoustic modes depend on the enclosure height, with decreasing modal frequencies as Lz increases and, due to the system degeneracy, several mode pairs have identical frequencies.
552
Fluid-structure interaction
Figure 6.43. Modal frequencies of the square membrane in vacuum (circles) and of the rigid enclosure (squares) as a function of the enclosure height
Figure 6.44. Modal frequencies of the vibroacoustic coupled system as a function of the enclosure height
Vibroacoustic coupling
553
Figure 6.44 displays the modal frequencies of the corresponding vibroacoustic coupled system, which depend on Lz in a complex manner. As stated before, timpani enclosures are designed so that several lower-frequency vibroacoustic resonances are tuned, as best as possible, according to “musically interesting” frequency ratios. Actually, the first anti-symmetric modes (with one, two and three diameter nodal lines) are typically tuned with frequency ratios 1:1.5 : 2 (a frequency ratio of 1.5 is a perfect fifth musical interval, and a ratio of 2 is an octave above the fundamental). The vibroacoustic modal frequencies presented in Figure 6.44 show significant changes with the enclosure size – both increasing and decreasing, depending on the mode – suggesting that such delicate tuning may be attempted. However, before going further, remember that the example presented here, for analytical simplicity, has not the circular membrane and dome-shaped enclosure found in orchestral timpani. Also, the membrane load associated with the external pressure field, which lowers significantly the modal frequencies, is not included in the present formulation. Therefore, as far as actual kettledrums are concerned, a detailed analysis of the vibroacoustic modes here computed would be pointless. The present example is however sufficient to capture the qualitative features of the vibroacoustic coupling mechanism which is used to tune the real instrument. Basically, Figure 6.44 shows the existence of a zero frequency coupled mode, and two types of vibroacoustic behaviour, according to whether the natural frequencies increase, or decrease, as Lz increases. Colour plates 10 and 11 display the decoupled membrane and enclosure modes, respectively, as well as those of the coupled system, for a low volume enclosure (with Lz = 0.2 m). Also, colour plates 12 and 13 show the corresponding results for a larger volume enclosure (with Lz = 0.7 m). It is clear, from the first mode shapes in colour plates 10 and 12, that the zero frequency coupled mode is connected with the first acoustic mode of the enclosure (constant pressure), which now enforces a constant deformation of the membrane, close to its first mode shape. The amplitude of the membrane deformation may be easily inferred from the static version of [6.205]: πx πy ⎧ ∂2Zs ∂2Zs 1 ⎪ Z s φ11 ( x, y ) a11 = sin L sin L a11 + 2 = p ( x, y ,0 ) with ⎨ x y ∂x 2 ∂y T ( p) ⎪ = = p x y b b ϕ ( , , 0) 1 ⎩ 000 000 000
[6.216]
Using the Rayleigh-Ritz or Galerkin procedure (see [AXI 05], Chapter 5), equation [6.216] leads to: Lx L y
∫∫ 0 0
⎛ 2 L2x + L2y ⎞ πx πx πy 1 πy − b000 ⎟ sin dxdy = 0 sin sin ⎜⎜ −π a11 2 2 sin ⎟ L L L L T L Ly x y x x y ⎝ ⎠
⇒
2 x
2 y
a11 16 L L =− 4 2 ≈ −5 10−6 b000 π T Lx + L2y
m N/m
2
[6.217]
554
Fluid-structure interaction
The approximate result [6.217] agrees well with the eigenvector components for the zero frequency coupled mode stemming from [6.215]. As could be expected, modes with decreasing frequencies when the container length increases are related to the symmetrical motions of the membrane, which implies an elastic effect due to the change in the acoustical volume. Stiffness provided by fluid elasticity obviously decreases as the container length is increased, as well as when the modal wavelengths are shortened, which means that the mode rank increases. Modes 4 (240 Hz) and 7 (380 Hz), when Lz = 0.2 m, and modes 2 (151 Hz) and 5 (289 Hz), when Lz = 0.7 m, are particularly suitable to illustrate such a general trend. Notice that the value 240 Hz is higher than the modal frequency in vacuum (179 Hz) of the relevant membrane mode shape, indicating the importance of the stiffening effect provided by the small acoustic volume at Lz = 0.2 m. When Lz = 0.7 m, the corresponding vibroacoustic modal frequency is lowered to 151 Hz, which is below the membrane uncoupled frequency, indicating thus that for large enclosures coupling becomes mostly inertial. Actually, for the intermediate value Lz ≈ 0.52 m, the coupled mode displays the same frequency as the membrane in vacuum, which is due to the fact that, at this frequency, the acoustical field of such enclosure is almost zero at the membrane interface. Finally, the acoustical load on all anti-symmetric modes of the membrane is of inertial nature, as typified by modes 2 or 3 (211 Hz) and mode 5 (301 Hz), when Lz = 0.2 m, and by modes 3 or 4 (225 Hz) and mode 6 (310 Hz), when Lz = 0.7 m. For such modes the fluid is hardly compressed, being mostly “sloshed” with the membrane motion. Such inertial effect is higher when the fluid is confined, decreasing for higher enclosure volumes, hence the observed increasing vibroacoustic modal frequencies. For the interested reader, a general introduction to drums and timpani can be found in [FLE 98]. These instruments have been studied by a number of researchers, including the earlier findings by Lord Rayleigh [RAY 94], see in particular [MOR 48], [DE 71], [BEN 76], [ROS 76] and [CHR 84]. 6.4.3
Vibroacoustic consequences of cavitation
6.4.3.1 One dimensional model of cavitation As outlined in Chapter 1, subsection 1.2.2.6, if the absolute pressure in a liquid tends to fall below a certain threshold value Pc , called pressure of cavitation, liquid vaporizes and pressure remains constant. As a consequence, a certain volume of vapour and dissolved gases grows as long as the fluid is stretched. When the cavitating liquid is compressed again, the vapour bubbles, or pockets, collapse very quickly, inducing strong short-lived pressure transients which excite violently the structures. A well known example is the so called water hammer phenomenon which can occur in piping systems, in particular house plumbing, as the result of cavitation due to negative pressure waves, see for instance the recent historical review by Bergant et al. [BER 06]. As illustrated in Chapter 5, subsection 5.3.5.3, such
Vibroacoustic coupling
555
negative pressure pulses can be induced either directly or after reflection by external exciting transients, such as the rapid opening or closure of a tap.
Figure 6.45. One dimensional model for cavitation
In a one-dimensional vibroacoustic analysis, cavitation can be modelled according to the phenomenological scenario shown in Figure 6.45. Let us assume that according to the linear calculation carried out in the elastic domain of fluid response, pressure is found to fall below Pc at a certain time t0 , and at some place x0 along the tube. It is thus appropriate to initiate a vaporization process at t0 and x0 by prescribing the necessary conditions to maintain the absolute pressure P ( x0 ; t0 ) at the constant value Pc . From the physical standpoint this is achieved by
the vaporization of a certain amount of liquid around the cross-section S f ( x0 ) . The cavitating volume occupied by the vapour is modelled as a gas column bounded conceptually by the cross-sections S f ( x1 ) and S f ( x2 ) , where x1 < x0 < x2 , which is consistent with the general simplifying assumptions of a one-dimensional model. Pressure is kept constant in the cavitating volume by feeding it with exactly the amount of liquid which corresponds to the liquid driven out, or in, by the motion of the boundaries S f ( x1 ) and S f ( x2 ) . Denoting q1 and q2 the corresponding volume velocities of liquid, the balance of liquid material fed into the cavitating volume is written as: Q ( t ) = q1 − q2
[6.218]
556
Fluid-structure interaction
The mass of liquid driven by the vapour column at time t follows: ⌠
t
M ( t ) = ρ f ⎮⎮ Q ( t ′) dt ′
[6.219]
⌡t0
where ρ f stands for the density of the liquid. Depending on the time variation of the pressure in the columns of liquid, the cavitating volumes are found to grow ( d M ( t ) / dt > 0 ) or alternatively to collapse
( d M ( t ) / dt < 0 ) .
If at some time tc > t0 , M ( tc ) vanishes, this means that the
cavitating volume is nil, fluid column being continuous again. However, at the time tc at which the cavitating volume collapses entirely, the volume velocities q1 and q2 at the interface with each adjacent column of liquid are generally not the same. As a consequence, the liquid columns are found to impact each other at tc given by
the conditions M ( tc ) = 0 and q1 ( tc ) ≠ q2 ( tc ) . After the impact, the liquid is again
continuous and in local equilibrium, which implies in particular that q1 = q2 . Therefore, the impact is modelled as a step function of volume velocity, which balances exactly the finite discontinuity Q ( tc ) due to cavitation at the time of the impact. The acoustic source associated with the step −Q ( tc ) U ( t - tc ) triggers a forward and a backward pressure waves at x0 and t = tc which can be determined analytically by solving the forced acoustic equation: ∂2 p 1 ∂2 p − =0 ∂ x 2 ce2 ∂ t 2 ∂p Sf ∂x
x0
∂p − Sf ∂x +
= ρ f Q ( tc ) δ ( t − tc )
[6.220]
x0 −
Actually, the backward and forward waves are identical to each other and only one of them needs to be obtained explicitly, the forward wave for instance, which is found to be: p ( x ≥ x0 ; t ≥ tc ) = −
ρ f ce Q ( tc ) Sf
⎛ ( x − x0 ) ⎞ U ⎜t ⎟ ce ⎠ ⎝
[6.221]
6.4.3.2 Analytical example It is found interesting to solidify the model described above by applying it to a problem sufficiently simple to be solved analytically and appropriate to illustrate the vibroacoustic effects of cavitation.
Vibroacoustic coupling
557
Figure 6.46. Tube closed at one end containing a liquid which may cavitate
As shown in Figure 6.46, the system analysed here is the same tube as that used in subsection 6.2.2.3. Again, the system is supposed at rest at times t ≤ 0 . The tube is assumed to be rigid, its mass is M s and to simplify further the problem, stiffness of the supports is neglected. The contained liquid is supposed to vaporize as soon as the absolute pressure tends to be negative. The system is excited by an external force concentrated at the closed bottom and in the longitudinal direction. The time history of the transient is specified in Figure 6.47.
Figure 6.47. Time history of the excitation force
As an essential feature, it comprises two successive steps of equal duration 2θ . During the first step, as the force is positive, the fluid column is stretched and the pressure is decreased, being minimal at x = L. Vaporization is assumed to start at a certain time t0 < 2θ . It is noted that due to the cavitating volume, the force acting on
the column of liquid remains constant and equal to S f ( P0 − Pc ) , where P0 is the static pressure at the tube inlet, supposed to be constant. The force acting on the solid is the external force plus the pressure component which is − S f ( P0 − Pc ) . As a particular case of academic interest, if P0 = Pc the fluid column moves as a free rigid body at the constant velocity X s ( t0 ) and as long as the external force remains positive the tube is accelerated, hence the cavitating volume grows. During the second step the exciting force is negative hence the tube is decelerated and the
558
Fluid-structure interaction
cavitating volume starts to collapse. At a certain time the tube and the column of liquid are impacting each other. The scenario remains qualitatively the same even if Pc differs from P0 as it is the case in reality. Mathematical analysis of the problem is facilitated if the total duration 4θ of the whole transient is made shorter than the back and forth travel time 2τ of the sound waves in the tube, which is the case if the following numerical values are selected: L = 10 m ; M S = 97 kg ; S f = 0.03 m 2 ; θ = 5 ms ; F0 = 2.5 kN ; α F0 = −10 kN ce = 500 ms -1 ; ρ f = 1 000 kgm -3 ; M f = 302 kg ; τ = L / ce = 20 ms ; Pc = −0.1 bar
Adopting such values, cavitation is initiated at 0 < t0 < θ and total collapse of the cavitating volume occurs at 3θ < tc < 4θ , as shown by the following analysis. Figure 6.48 is of the same nature as Figure 6.45; it indicates that the problem can be treated analytically as a sequence of four distinct steps, each one being described by linear equations. 1. Initial non cavitating step: 0 ≤ t ≤ t0 The system is governed by the following equations: e M s X1 = S f p ( L; t ) + F ( ) ( t )
∂2p 1 ∂2p − =0 ∂ x 2 ce2 ∂ t 2 ∂p ∂x
p ( 0; t ) = 0 ;
[6.222] = − ρ f X s x=L
Laplace transforming [6.222] gives: S p ( L; s ) F ( e ) ( s ) X 1 ( s ) = f + ; M s s2 M s s2
⎛ 1 − e −τ s ⎞ p ( L; s ) = − ρ f ce sX 1 ( s ) ⎜ −τ s ⎟ ⎝ 1+ e ⎠
[6.223]
whence: e F ( ) ( s ) ⎛ ⎞ 1 − e −τ s X 1 ( s ) = ⎜ ⎟ M s s ⎝ s + ω0ς va + ( s − ω0ς va )e −τ s ⎠ e ω ς F ( ) ( s ) ⎛ e −0.5τ s (1+ x / L ) − e −0.5τ s (1− x / L ) ⎞ −τ s p ( x; s ) = 0 va ⎜⎜ −τ s ⎟ ⎟ (1 + e ) ω ς ω ς ( ) Sf s s e + + − 0 0 va va ⎝ ⎠
[6.224]
where: ωva = ω0ς va =
μf τ
F e ; F ( ) = 12 θs
[6.225]
Vibroacoustic coupling
559
Figure 6.48. Cavitation initiated at the closed end of a moving tube
The retarded terms related to wave reflections can be discarded since the whole transient lasts 2τ . Inverse Laplace transforms of [6.224] follow as: F1 p ( L; t ) = − ( e−ω0ς vat + ω0ς va t − 1) θω0ς va S f ⎛ (ω0ς va t )2 ⎞ − ω0ς va t − e −ω0ς va t + 1⎟ ⎜ 3 ⎟ 2 M sθ (ω0ς va ) ⎜⎝ ⎠ F1 X 1 (t ) = e −ω0ς va t + ω0ς va t − 1) 2 ( M sθ (ω0ς va ) F1
X 1 (t ) =
[6.226]
Solutions [6.226] are used to determine the dynamical state of the system at the end of the first step, which are used as initial conditions to describe the second step. The following numerical values are found: t = 0.3 ms ; X = X ( t ) = 0.2 mm ; X ( t ) = X = 2 ms-1 0
0
1
0
1
0
0
2. Cavitating step: t0 ≤ t ≤ tc During this step, the solid is disconnected from the liquid and its motion is governed by the following equations: M s X 2 = F ( e ) ( t ) + ( Pc − P0 ) S f [6.227] X ( t ) = X ; X ( t ) = X 2
0
0
2
0
0
560
Fluid-structure interaction
Displacement of the tube is found to be:
(P − P ) S 1 2 X 2 ( t ) = X 0 + X 0 ( t − t0 ) + c 0 f ( t − t0 ) + Ms 2M s
t
⌠ ⎮ ⎮ ⌡t0
t′
dt '
⌠ ⎮ ⎮ ⌡t0
F(
e)
( t ′′ ) dt ′′
[6.228]
Adopting the external force depicted in Figure 4.47, the displacement law [6.228] is made explicit as: t0 ≤ t ≤ θ
F(
;
e)
(t ) =
F0 t θ
P ( t − t0 ) F0 X 2 ( t ) = X 0 + ( t − t0 ) X 0 + c + ( t 3 + 2t03 − 3tt02 ) 2M s 6 M sθ 2
[6.229]
P ( t − t0 ) F0 X 2 ( t ) = X 0 + c + ( t 2 − t02 ) 2 M sθ Ms θ ≤ t ≤ 2θ
F(
;
e)
(t ) =
F0 ( 2θ − t ) θ
P (t − θ ) + X 3 ( t ) = X θ + ( t − θ ) X θ + c 2M s
F0 ( 2θ 3 + 3tθ 2 − t 3 )
P (t − θ ) − X 3 ( t ) = X θ + c Ms
2 F0 ( t − θ ) Ms
2
2θ ≤ t ≤ 3θ
;
F(
e)
F0 ( t 2 − θ 2 )
(t ) =
2 M sθ
6 M sθ
+
F (t − θ ) + 0 Ms
2
[6.230]
4 F0 ( 2θ − t ) θ
X 4 ( t ) = X 2θ + ( t − 2θ ) X 2θ +
2 2 3 2 3 Pc ( t − 2θ ) 2 F0 ( t − 12tθ + 16θ ) 4 F0 ( t − θ ) − + 2M s 3M sθ Ms
2 F0 ( t 2 − 4θ 2 )
P ( t − 2θ ) − X 4 ( t ) = X 2θ + c Ms
M sθ
+
8F0 ( t − 2θ ) Ms
[6.231] 3θ ≤ t ≤ tc
;
F(
e)
(t ) = −
4 F0 ( 4θ − t ) θ
P ( t − 3θ ) + X 5 ( t ) = X 3θ + ( t − 3θ ) X 3θ + c 2M s 2
P ( t − 3θ ) + X 5 ( t ) = X 3θ + c Ms
2 F0 ( t 3 + 54θ 3 − 27θ 2 t ) 8F0 ( t − 3θ )2 − 3M sθ Ms
2 F0 ( t 2 − 9θ 2 ) 16 F0 ( t − 3θ ) − M sθ Ms
[6.232]
Vibroacoustic coupling
561
On the other hand, it can be claimed that the interface separating the liquid and the vapour remains essentially at the velocity X 0 . Indeed, the pressure at and near the interface is practically unchanged as cavitation is initiated and the action of the force S f ( P0 − Pc ) related to the pressure unbalance between the tube inlet and the cavitating fluid can be neglected in the time scale of the transient because it is much shorter than the acoustic travel time. Thus, displacement of the liquid-vapour interface is approximated as: X ( t ) = X + X ( t − t ) [6.233] f
0
0
0
3. Collapse: t = tc The amplitude of the pressure wave induced by the impact between the solid and the liquid is much greater than Pc and the force imparted to the tube is much larger than the external force which is nearly zero at the time the vapour column collapses entirely. Hence, the motion of the tube, before any back reflection of sound waves, is practically governed by the equation:
(M
s
+ M f ) X6 = − PS f U (t − tc ) = ρ f ceQ ( tc ) U (t − tc )
X 6 ( tc ) = X 5 ( tc ) ;
X 6 ( tc ) = X 4 ( tc )
[6.234]
The pressure step induced by the collapse is P 1.6 bar . With the aid of relation [6.221] it is expressed in terms of the volume velocity discontinuity Q ( tc ) just before the collapse. The motion of the tube and the reconnected fluid is: X 6 ( t ) = X 5 ( tc ) + X 5 ( tc )( t − tc ) + X 6 ( t ) = X 5 ( tc ) +
PS f ( t − tc )
PS f ( t − tc )
2(Ms + M f
2
)
[6.235]
Ms + M f
The analytical results are illustrated in Figure 6.49 to 6.51. Figure 6.49 is a plot of the pressure field at the closed end of the tube. Three distinct successive steps can be clearly recognized, namely an initial drop corresponding to the precavitating step, followed by a flat valley corresponding to the cavitating step which ends at the time of collapse t = tc at which the pressure rises abruptly as a step function and is maintained essentially constant until the first reflected wave reaches the closed end of the tube. Figure 6.50 is a plot of the tube displacement (full line) and that of the liquid-vapour interface. The differential motion of the solid and the liquid during the cavitating step is clearly visible. The effect of the pressure step induced by the collapse is to decelerate the back motion of the tube. However, this effect can be brought in evidence much better by looking at the velocity plots of Figure 6.51
562
Fluid-structure interaction
rather than at the displacement plots. At the time of the collapse, the sign of the fluid velocity is changed to reach a common value with the solid. Due to the impact with the fluid, the acceleration of the tube is subjected to a finite discontinuity as attested by the abrupt change in the slope of the velocity curve.
Figure 6.49. Time history of the pressure at the closed end of the tube
Figure 6.50. Displacements of the tube and the liquid
Vibroacoustic coupling
563
Figure 6.51. Velocity of the tube and the liquid-vapour interface
6.5. Finite element method 6.5.1
Introduction
In industrial applications, recourse to numerical tools is necessary to analyse acoustic and vibroacoustic responses of complicated systems. As mentioned earlier, in practice most of them are based on either finite element methods or on boundary element methods, mixing both techniques eventually. In principle at least, use of either BEM or FEM can be recommended according to whether the material system (fluid or solid) extends to infinity or not. However, the choice also depends on the taste and professional skills of the user, since numerical techniques are available nowadays which allow one to deal with closed and open space domains by using one or the other of the BEM or FEM methods. FEM is implemented in several large purpose computer programs such as NASTRAN, ANSYS, CASTEM etc., and is used in various fields of application, provided the extent of the fluid domain to be meshed is not too large. In this respect it is worth mentioning that even infinite volumes of fluid can often be meshed by using a limited collection of finite elements, provided absorbing elements are made available to simulate anechoic boundaries (see Chapter 7, subsection 7.1.3.2). An introductory description of the FEM emphasizing the most salient features of the discretization procedure can be found in [AXI 05] in the context of structural mechanics. For mathematical convenience straight beams and beam assemblies were used as illustrative examples. The aim here is to outline the extension of the method to fluid-structure coupled systems, focusing on the symmetrical ( X s , Π , p ) formalism. However, in a book of this nature there is no place to enter deeply in the subject which presents many numerical and computational subtleties. The reader interested in this vast and rapidly evolving subject is referred to the specialized literature, see in particular [ZIE 89],
564
Fluid-structure interaction
[EVE 81], [MOR 95], [MAK 99a,b], [SIG 05], [SIG 06]. Presentation here is drastically restricted to the most fundamental aspects which concern the energy functionals to be discretized and the generic features of the matrix equations which form the finite element model of the vibroacoustic problem, focusing on the fluid and the fluid-structure coupling part of the model. 6.5.2
Variational formulation of the vibroacoustic equations
The formalism established in section 6.3 in the particular context of plane waves in tubular circuits is extended to the three dimensional case by using the variational approach, which serves as the starting point to discretize the continuous problem in finite elements. As already emphasized in [AXI 05] Chapter 3, the method can be understood as an application of the weighted integral formulation of the continuous elastodynamics. Once more, the finite volumes occupied by the fluid and the solid are denoted (Vf ) and (Vs ) , respectively. The fluid-structure interface, or wetted wall, is denoted (W ) and finally (S f
)
stands for the surface bounding the finite
volume (Vf ) . 6.5.2.1 Formulation in terms of fluid displacement The dynamic equilibrium of the coupled system is described first by using the local equations, where the fluid-structure coupling is treated as an elastic connection sufficiently stiff to produce a satisfactory approximation of the holonomic constraint which relates the displacement of the fluid and that of the structure at every location on (W ) , which reads as: X s − X f .nδ ( r − r0 ) = 0; ∀ r0 ∈ (W ) [6.236]
(
)
Accordingly, the local equations which govern the motion of the coupled system are written as: K s ⎡⎣ X s ⎤⎦ + M s ⎡⎢ X s ⎤⎥ + K L n ( X s − X f ) .nδ ( r − r0 ) = F ( e ) ( r ; t ) ∀ r ∈ (Vs ∪W ) ⎣ ⎦ −grad ( ρ f ce2 divX f ) + ρ f X f − K L n ( X s − X f ) .nδ ( r − r0 ) = S ( e ) ( r ; t ) ∀ r ∈ (Vf ∪W ) [6.237] where K L is the stiffness coefficient of the spring which is supposed to connect the fluid to the structure in the normal direction n and at every position r0 on
(W ) (W ) .
The support conditions of the structure are tacitly assumed to be
included in the stiffness operator K s [
]
and possibly also in the mass operator
M s [ ] , where again the brackets are used to mark the difference between a scalar
coefficient and an operator. The boundary condition fulfilled by the fluid at every point on (S f ) ∩ (W ) is assumed to be a conservative impedance, expressed here as:
Vibroacoustic coupling
a grad p.n + bp = 0 ⇒ a grad E f div X f .n + b E f div X f = 0; ∀r ∈ (S f ∩W )
(
(
(
)
))
(
565
[6.238]
)
where it is supposed that no external sources are present at the boundaries to alleviate the formalism, though there would be no difficulty to include them. Furthermore, in practice the boundary condition [6.238] can be further simplified for the two following reasons. The first reason is that in most applications Young’s modulus of the fluid is constant, on the boundary or on significant parts of (S f ) , and similarly for the surface impedances. The second reason is that here the fluid motions we are interested in are irrotational. Assuming thus to simplify the presentation that the fluid properties and surface impedances are uniform on (S f ) , relations [6.238] can be replaced by the simpler expression: aΔX f .n + bdivX f = 0; ∀r ∈ (S f ∩W )
[6.239]
The standard boundary conditions are recovered by assuming that either b, or a is zero, the first case corresponding to a fixed boundary and the second to a free boundary. The Lagrangian of the coupled system is of the generic type: Lva = Ls + Lf + Lsf
[6.240]
In order to obtain the energy functionals entering into Lva , it is appropriate to use X s and X f as weighting fields to transform the local equations [6.237] into the weighted integral equations: Lva = Ls + Lf + Lsf = X s , F (e) ( r ; t ) X s , K s ⎡⎣ X s ⎤⎦ + M s ⎡⎢ X s ⎤⎥ + K L n X s − X f .nδ ( r − r0 ) ⎣ ⎦ (VS ) (VS ∪W ) = X s ,S ( e ) ( r ; t ) X f , −grad E f divX f + ρ f X f − K L n X s − X f .nδ ( r − r0 )
(
(
(
)
)
(
)
)(
V f ∪W
)
(Vf )
[6.241] where U .V stands for the scalar product in the Euclidean space and U ,V for (D ) the functional scalar product of U .V in the domain (D ) in agreement with the notations introduced in [AXI 05]. As explicitly specified, the domain of integration includes both the solid and fluid volumes and the fluid-solid interface. The weighted integrals [6.241] bring in evidence the following energy functionals:
566
Fluid-structure interaction
1. Functional of structural elasticity Ees = X s , K s ⎡⎣ X s ⎤⎦
(
⌠
)(
Vs ∪W )
= ⎮⎮
⎮ ⌡(Vs )
X s . K s ⎡⎣ X s ⎤⎦ dVs
)
(
[6.242]
2. Functional of structural inertia Eis = X s ,⎛⎜ M s ⎡ X s ⎤ ⎞⎟ ⎢⎣ ⎥⎦ ⎠ ⎝
⌠ ⎮
(Vs ∪W )
=⎮
⎮ ⎮ ⌡(Vs )
X s . ⎛⎜ M s ⎡ X s ⎤ ⎞⎟ dVs ⎢⎣ ⎥⎦ ⎠ ⎝
[6.243]
3. Functional of structure-fluid coupling: Esf = X s , K L n X s − X f .nδ ( r − r0 )
(
)
⌠ ⎮
(Vs ∪W )
(( X .n ) − ( X .n )( X .n )) dW
= KL ⎮
⎮ ⌡(W )
2
s
s
f
[6.244] 4. Functional of fluid elasticity: Eef = X f , −grad E f divX f
(
)(
⌠
Vf ∪W
)
= ⎮⎮
⎮ ⌡(Vf
)
− X f .grad E f divX f dVf
(
)
[6.245]
5. Functional of fluid inertia: Eif = X f , ρ f X f
⌠ = ⎮⎮ ρ f X f .X f dVf (Vf ∪W ) ⎮⌡(Vf )
[6.246]
6. Functional of fluid-structure coupling: E fs = X f , K L n X f − X s .nδ ( r − r0 )
(
)
⌠ ⎮
(Vs ∪W )
= KL ⎮
⎮ ⌡(W )
(( X .n ) − ( X .n )( X .n )) dW 2
f
s
f
[6.247] 7. Functional of external excitation of the structure: e EFs = X s , F ( ) ( r ; t )
⌠
(Vs ∪Ss )
= ⎮⎮
⌡(Vs )
e X s .F ( ) ( r ; t ) dVs
[6.248]
Vibroacoustic coupling
567
8. Functional of external excitation of the fluid: Eef = X f , S ( e ) ( r ; t )
⌠
(Vf )
= ⎮⎮
⌡(Vf
)
X f .S ( e ) ( r ; t ) dVf
[6.249]
As illustrated in many examples in [AXI 05] in the context of the elastodynamics of solids, by integrating judiciously the volume functionals by parts, either with respect to time for the inertia terms, or with respect to space for the elastic terms, they can be transformed into quadratic, symmetric and positive forms. It is of special interest to detail the calculation for the elastic functional of the fluid. As for the boundary conditions [6.239], it is sufficient to consider the case of a uniform fluid and because curl X f = 0 , the functional [6.245] simplifies into:
( )
Eef = X f , −grad E f divX f
(
)(
Vf ∪W
⌠
)
= E f ⎮⎮
⌡(Vf )
− X f .ΔX f dVf
[6.250]
An integration by parts leads to the following integrals: ⎧⌠ ⌠ ⎪ − X f .ΔX f dVf = E f ⎨ ⎮⎮ − divX f X f .n d S f + ⎮⎮ divX f ⎮ ⎮ ⌡(Vf ) ⌡(Vf ) ⎪⎩ ⌡(S f ) ⌠
(
Eef = E f ⎮⎮
)
(
)
2
⎫ ⎪ dS f ⎬ ⎪⎭ [6.251]
It is noted that for the standard boundary conditions the surface integral vanishes in the domain (S f ∩W ) . On the other hand, recalling that the fluctuating pressure is precisely p = − E f divX f and that on the wetted wall the condition X f .n = X s .n is verified, the surface functional in [6.251], restricted to (W ) , identifies with that of the structure-fluid coupling term: ⌠
Esf = − E f ⎮⎮
⎮ ⌡(W )
⌠ ⎮ f .n dW = ⎮
( divX ) X f
⌡(W )
pX s .n dW
[6.252]
However, the contribution of this term must be discarded since in the spring model adopted here, the corresponding energy is already accounted for by the functional [6.247]. Before leaving the subject of the energy functionals pertinent to the X s , X f
(
)
formalism, it is also useful to outline the model constrained by the holonomic condition which relates the displacement of the fluid and that of the structure at every location on (W ) . According to the constrained model, the local equations are written as follows:
568
Fluid-structure interaction
e K s ⎣⎡ X s ⎦⎤ + M s ⎡⎢ X s ⎤⎥ = F ( ) ( r ; t ) ∀ r ∈ (Vs ) ⎣ ⎦ e − grad E f divX f + ρ f X f = S ( ) ( r ; t ) X s − X f .nδ ( r − r0 ) = 0 ∀r ∈ (Vf ) ; ∀ r0 ∈ (W )
(
(
)
)
[6.253]
Once more the Lagrange formalism is revealed as an elegant and powerful method to treat such equations. The Lagrangian of the constrained system reads as: ⌠
Lva = L ′ = LS + Lf + Ls′f = Ls + Lf + ⎮⎮
⎮ ⌡(W )
Λ ( r0 ) X s − X f .n dW
(
)
[6.254]
As in [6.240], Ls and Lf are the uncoupled Lagrangians of the solid and fluid, respectively. Fluid-structure interaction is entirely accounted for by the functional Ls′f , where Λ ( r0 ) stands for the Lagrange multiplier field associated with the holonomic constraint. 6.5.2.2 Mixed X S , p formulation
(
)
We start from the local equations written in terms of the mixed variables X s and p as: K s ⎡⎣ X s ⎤⎦ + M s ⎡⎢ X s ⎤⎥ + pnδ ( r − r0 ) = F ( e ) ( r ; t ) ∀ r ∈ (Vs ) ; ∀ r0 ∈ (W ) ⎣ ⎦ [6.255] 1 e p + ρ f X s .nδ ( r − r0 ) = S ( ) ( r ; t ) ; ∀ r ∈ (Vf ) ; ∀ r0 ∈ (W ) Δp − 2 ce
In order to obtain the energy functionals entering into the problem, it is appropriate to use the fields X s and p as weighting functions to transform the local equations [6.255] into the weighted integrals: = X s , F (e) ( r ; t ) X s , K s ⎡⎣ X s ⎤⎦ + M s ⎡⎢ X s ⎤⎥ + pnδ ( r − r0 ) ⎣ ⎦ (Vs ∪W ) (V ∪W )
(
)
s
⎞ ⎛ 1 p, ⎜ Δp − 2 p + ρ f X s .nδ ( r − r0 ) ⎟ = p, S ce ⎝ ⎠ (Vf ∪W )
(e)
[6.256]
(r;t )
(Vf ∪W )
The weighted integrals [6.256] lead one to define the following new expressions for the fluid and vibroacoustic functionals: 1. Functional of structure-fluid coupling: Es′f = p, X s .nδ ( r − r0 )
(
)(
Vs ∪W )
⌠
= ⎮⎮
⌡(W )
pX s .n dW
[6.257]
Vibroacoustic coupling
569
2. Functional of fluid elasticity: ⌠
⎮ 1 pp Eef′ = p, 2 =⎮ p dVf ⎮ ce c2 (Vf ∪W ) ⎮⌡(V ) e f
[6.258]
3. Functional of fluid inertia: ⌠
Eif′ = p, Δp (V ∪W ) = ⎮⎮ pΔp dVf f ⌡(Vf )
[6.259]
4. Functional of fluid-structure coupling: E f′s = p, ρ f X s .nδ ( r − r0 )
(
)
⌠
= ⎮⎮
⎮ ⌡(W )
(Vs ∪W )
ρ f pX s .n dW
[6.260]
5. Functional of external excitation of the fluid: Eef′ = p, S
(e)
(r;t)
⌠
= ⎮⎮
(Vf ∪S f )
⌡(Ve )
pS
(e)
( r ; t ) dVf
[6.261]
The functional of fluid inertia can be further transformed by using the vector identities: pΔp = p div grad p ; div(uV ) = V .gradu + u div V [6.262] With the aid of [6.262] it follows that: p div grad p = div p grad p − grad p
(
)
(
) (
)
2
[6.263]
Substituting [6.263] into [6.259] and using the integral transformation theorem for a divergence, we arrive at: ⌠ ⎮ ⎮ ⌡(Vf
⌠
)
pΔp dVf = ⎮⎮
⎮ ⌡(S f ∩W )
⌠ p grad p.n dS f − ⎮⎮
(
)
As for the elastic functional in the
⎮ ⌡(Vf )
( grad p ) dV
f
(X ,X ) s
2
f
[6.264]
formalism, the surface term is
restricted to the boundary other than the wetted wall, since the corresponding term is already accounted for in the fluid-structure coupling functional [6.260]. 6.5.2.3 Mixed X s , Π , p formulation
(
)
With the aid of relation [6.6], the local equations [6.255] are transformed into:
570
Fluid-structure interaction
e K s ⎡⎣ X s ⎤⎦ + M s ⎡⎢ X s ⎤⎥ + Π nδ ( r − r0 ) = F ( ) ( r ; t ) ∀ r ∈ (Vs ) ; ∀ r0 ∈ (W ) ⎣ ⎦ p 1 (e) ΔΠ S ( r ; t ) ∀ r ∈ (Vf ) ; ∀ r0 ∈ (W ) − + X s .nδ ( r − r0 ) = 2 ρf ρ f ce ρf
[6.265]
p Π − =0 2 ρ f ce ρ f ce2
Using the fields X s , Π and p as weighting functions, the local system [6.265] is transformed into the weighted integrals: e X s , K s ⎡⎣ X s ⎤⎦ + M s ⎡⎢ X s ⎤⎥ + Π nδ ( r − r0 ) = X s , F ( ) (r;t ) ⎣ ⎦ (Vs ∪W ) V ∪W
(
)
⎛ ΔΠ ⎞ p .nδ ( r − r ) ⎟ X + + Π ,⎜ 0 ⎟ s 2 ⎜ ρ ρ f ce ⎝ f ⎠
(
s
)
= Π, (Vf ∪W )
1 S ρf
(e)
(r;t )
[6.266] (Vf ∪W )
⎛ p Π ⎞ p, ⎜ − =0 2 2 ⎟ ⎜ρ c ⎟ ⎝ f e ρ f ce ⎠ (Vf ∪W )
The functionals describing the fluid and the vibroacoustic coupling are as follows: 1 p Eef′′ = Π , ρ f ce2 ΔΠ Eif′′ = Π , ρf ⌠
Esf′′ = ⎮⎮
⌡(W )
= (Vf ∪W )
= (Vf ∪W )
⌠ ⎮ ⎮ ⎮ ⎮ ⌡(Vf
⌠ pX s .n dW = ⎮⎮
⌡(W )
E fs′′ = Π , X s .nδ ( r − r0 )
(
⌠ ⎮ ⎮ ⎮ ⎮ ⌡(Vf
)
)
Π p dVf ρ f ce2
[6.267]
ΠΔΠ dVf ρf
[6.268]
Π X s .ndW
[6.269]
)
⌠
(Vs ∪W )
= ⎮⎮
⌡(W )
Π X s .n dW
[6.270]
Two additional functionals arise due to the holonomic relation between Π and p which forms the third equation of the system [6.265]: ⌠
⎮ 1 p2 ′′ = p, 2 p dVf = ⎮⎮ EeH ce ρ f ce2 (Vf ∪W ) ⎮⌡ V
( f)
[6.271]
Vibroacoustic coupling
EiH′′ = p,
6.5.3
− Π ρ f ce2
= (Vf ∪W )
⌠ ⎮ − ⎮⎮ ⎮ ⌡(Vf
)
pΠ dVf ρ f ce2
571
[6.272]
Discretization in finite elements
As explained in more detail in [AXI 05], the geometrical domain occupied by the es solid is discretized into a set of finite parts, the number of which is denoted N ( ) , which defines the mesh of the finite element model as far as the structural part of the problem is concerned. The boundary of the solid (Ss ) is also discretized into a set of finite parts, the number of which is denoted N ( ) and it is appropriate to distinguish between those parts (W ) where fluid-structure interaction takes place, and the other bs
parts, where it does not. The functionals of the structure are approximated by summing the element functionals associated with every element of the discrete
(
model which occupies the space domain Vn ( Ees = X s , K s ⎡⎣ X s ⎤⎦ Eis = X s , M s X s
(Vs )
∑ n =1
(Vs ∪W )
e Ees = X s , F ( ) ( r ; t )
es N( )
es N( )
∑ n =1
(Vs ∪Ss )
X s , K s ⎡⎣ X s ⎤⎦ X s , M s ⎡⎢ X s ⎤⎥ ⎣ ⎦ es N( )
∑ n =1
):
es )
(V ( ) ) n
es
[6.273]
(V ( ) ) n
es
e X s , F ( ) (r;t )
(V ( ) ) n
es
The mesh is generated automatically by using geometrical supports of simple 1D, 2D and 3D shapes. Already at this step an approximation is likely to be made since the mesh does not map exactly the actual domain of the structure; whence the symbol used in [6.273]. On the other hand, the elementary functionals are calculated by using a set of suitable analytical approximations for the components of X s , which results in an interpolated displacement field denoted Ψ n ( r ; t ) , used to approximate the functionals of the n-th finite element as: n Ee(s ) = X s , K s ⎡⎣ X s ⎤⎦ (es ) Ψ n ( r ) , K s ⎡⎣Ψ n ( r ) ⎤⎦ (es ) (Vn ) (Vn ) Eis( n ) = X s , M s X s (es ) Ψ n ( r ) , M sΨn ( r ) es
(V )
(V ( ) )
n
e n EFs( ) = X s , F ( ) ( r ; t )
(
es Vn( )
[6.274]
n
)
e Ψ n (r ), F ( ) (r;t)
(V ( ) ) n
es
To perform the integrals involved in [6.274] it is convenient to use low degree polynomials of space coordinates to define the coefficients of Ψ n ( r ; t ) . The degree
572
Fluid-structure interaction
depends on the order of the differential equation to be solved and on the number of internal nodes used in the definition of the finite element, see for instance [BAT 82], [ZIE 89]. The coefficients are adjusted in such a way that the interpolated field fits the so-called nodal displacements X n ( t ) at each node of the element. As a result Ψ n ( r ; t ) is found to be a linear form of the nodal displacements of the element, which may be written as: ⎡⎣Ψ n ( r ; t ) ⎤⎦ = ⎣⎡ N n ( r ) ⎦⎤ ⎣⎡ X n ( t ) ⎦⎤
[6.275]
The discretization process concerns the space variables only and time can be either omitted, as it is the case in [AXI 05], or considered to be fixed, as it is the case here. The result of the discretization is to transform the elastic and inertia functionals into quadratic forms which can be written in matrix notation as: Ees [ X s ] [ K s ][ X s ] T
;
Eis [ X s ] [ M s ] ⎡⎣ X s ⎤⎦ T
[6.276]
where [ X s ] is the vector of the nodal displacements. Both of them are symmetric. Depending on the support conditions, the stiffness matrix
[ Ks ]
is positive, or
positive definite and the mass matrix [ M s ] is definite positive. On the other hand, the functional of external excitation of the structure is transformed into the linear form: ⌠
EFs = ⎮⎮
⌡(Vs )
e T e X s . F ( ) ( r ; t ) dVs ≈ [ X s ] ⎡⎣ F ( ) ⎤⎦
[6.277]
Applying a variational principle such as Hamilton’s principle of least action or the principle of virtual work, leads to the equations of the finite element model which in the case of a forced problem described in the time domain have the following canonical form:
[ K s ][ X s ] + [ M s ] ⎡⎣ Xs ⎤⎦ = ⎡⎣ Fs(e) ( t )⎤⎦
[6.278]
Denoting NDs the number of degrees of freedom of the finite element model which
describes the solid, the size of the [ K s ] and [ M s ] matrices is NDs × NDs while the e column vectors [ X s ] and ⎡⎣ Fs( ) ⎤⎦ of the unknown displacement vector and external load respectively have NDs components.
As far as the fluid and vibroacoustic coupling functionals are concerned, the discretization process of the energy functionals proceeds along the same lines as for the solid. The geometrical domain occupied by the fluid is discretized into a set of ef finite parts, the number of which is denoted N ( ) . The boundary (S f ) is also discretized into a set of finite parts and it is appropriate to distinguish between those
Vibroacoustic coupling
573
parts where fluid-structure interaction takes place, and the other parts such as a fixed wall or a water level. The functionals relative to the whole domain occupied by the fluid are approximated by summing the element functionals associated with every element of the discrete model, using appropriate interpolating functions to describe the pressure and the Π field within the element. This leads to the definition of various matrices which are named using the same notations as in the matrix equations [6.150], [6.163] and [6.174], which resulted from the modal discretization of the same operators as here. Furthermore, it is recalled that the matrices are qualified to be either of the [ K ] or [ M ] type according to the mathematical analogy with the canonical form [6.278] and not to the physical nature of the fluid terms. Denoting ND f the number of DOFs of the finite element model describing the e fluid, the column vectors [ X va ] and ⎡⎣ Fva( ) ⎤⎦ have NDva = NDs + ND f components. Hereafter, the components are ordered by considering first the DOFs of the solid and then those of the fluid. Whatever the nature of the discretized equations may be, the fluid is described by a pair of square ND f × ND f matrices while the vibroacoustic
coupling is described by a pair of rectangular matrices, of size NDs × ND f and ND f × NDS . Of course, the constitutive matrices of the finite element model depend
on the nature of the equations which are discretized, as described below, considering the X s , X f , X s , p and X s , Π , p formalisms, successively.
(
) (
)
(
)
6.5.3.1 Finite element equations in the X s , X f
(
) variables
By discretizing the functionals [6.242] to [6.249], the following fluid and vibroacoustic finite element matrices, are defined:
⌠
Eef = ρ f ce2 ⎮⎮
⎮ ⌡(Vf )
)
( ) dV
⌠ ⎮
Esf = K L ⎮
f
X f
⌠ ⎮
Eif = ρ f ⎮
⎮ ⎮ ⌡(Vf
( divX )
⎮ ⌡(W )
2
f
2
T
dVf = ⎡⎣ X f ⎤⎦ ⎡⎣ K f ⎤⎦ ⎡⎣ X f ⎤⎦
T ⎡⎣ X f ⎤⎦ ⎡⎣ M f ⎤⎦ ⎡⎣ X f ⎤⎦
(( X .n ) − ( X .n )( X .n )) dW
2
s
s
T
f
T
K L ⎣⎡ X s ⎦⎤ ⎡⎣ K sf ⎤⎦ ⎣⎡ X s ⎦⎤ − K L ⎣⎡ X s ⎦⎤ ⎡⎣ K sf ⎤⎦ ⎣⎡ X f ⎦⎤ E fs = X f , K L n X f − X s .nδ ( r − r0 )
(
)
T
(Vs ∪W ) T
K L ⎡⎣ X f ⎤⎦ ⎡⎣ K sf ⎤⎦ ⎡⎣ X s ⎤⎦ − K L ⎡⎣ X f ⎤⎦ ⎡⎣ K sf ⎤⎦ ⎡⎣ X f ⎤⎦
[6.279]
574
Fluid-structure interaction
Using Hamilton’s principle, or the virtual work principle, and transposing the E fs functional, it is not difficult to arrive at a finite element model of the generic type: ⎡ ⎡[ K s ] + ⎡ K sf ⎤ ⎤ ⎤ − ⎣⎡ K sf ⎦⎤ ⎡ ⎡ X ⎤ ⎤ ⎡ ⎡ F ( e ) ⎦⎤ ⎤ ⎣ ⎦⎦ ⎢⎣ ⎥ ⎡ [ X s ] ⎤ + ⎡[ M s ] [ 0] ⎤ ⎢ ⎣ s ⎦ ⎥ = ⎢ ⎣ ⎥ ⎢ ⎥ ⎢ ⎥ T ⎢ ⎡ ⎡ K ⎤ + ⎡ K ⎤T ⎤ ⎥ ⎢ ⎣⎡ X f ⎦⎤ ⎥ ⎢ [ 0] ⎣⎡ M f ⎦⎤ ⎥ ⎢ ⎡ X f ⎤ ⎥ ⎢ ⎡ S ( e ) ⎤ ⎥ ⎡ ⎤ − K ⎦ ⎣ ⎦ ⎣ ⎣ ⎦ ⎦ ⎢⎣ ⎣ ⎣ sf ⎦ ⎢⎣ ⎢⎣ ⎣ f ⎦ ⎣ sf ⎦ ⎥⎦ ⎥⎦ ⎣ ⎦ ⎥⎦ [6.280]
where the stiffness and mass matrices have the same standard properties as [ K s ] and
[ M s ] respectively.
As pointed out in [SIG 05] and [SIG 06], it is of interest from a pedagogical point of view at least, to analyse the asymptotic behaviour of the model [6.280] when fluid compressibility is discarded, which means that divX f = 0 . Hence ⎡⎣ K f ⎤⎦ also vanishes as the elastic energy functional Eef does. As a short exercise, let us set ⎡⎣ K f ⎤⎦ to zero in the modal problem. The system [6.280] is thus written as: ⎡ ⎡ ⎡[ K ] + ⎡ K ⎤ ⎤ − ⎡ K ⎤ ⎤ ⎡[ M s ] [ 0] ⎤ ⎤ ⎡ [ X s ] ⎤ ⎡[0]⎤ ⎣ sf ⎦ ⎥ 2 ⎢ ⎢ ⎣ s ⎣ sf ⎦ ⎦ − ω ⎢ ⎥⎥ ⎢ ⎥= T T ⎥ ⎢⎢ ⎥ ⎢ ⎡ X f ⎤ ⎥ ⎢⎣[0]⎥⎦ ⎡ ⎤ M 0 [ ] ⎢ f ⎦⎥ ⎡ ⎤ ⎡ ⎤ K K − ⎣ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ sf ⎦ ⎣ sf ⎦ ⎦ ⎣⎢ ⎣ ⎦⎥
[6.281]
The first line of the system reads as:
[ K s ][ X s ] + ⎡⎣ K sf ⎤⎦ ⎡⎣[ X s ] − ⎡⎣ X f ⎤⎦ ⎤⎦ − ω 2 [ M s ][ X s ] = 0
[6.282]
The second row implies: ⎡⎣ K sf ⎤⎦ ⎡⎣[ X s ] − ⎡⎣ X f ⎤⎦ ⎤⎦ = −ω 2 ⎡⎣ M f ⎤⎦ ⎡⎣ X f ⎤⎦
[6.283]
By substituting [6.283] into [6.282] the following modal equation is obtained:
[ K s ][ X s ] − ω 2 ⎡⎣[ M s ] + ⎡⎣ M f ⎤⎦ ⎤⎦ [ X s ] = 0
[6.284]
The result [6.284] is remarkable as it provides an independent proof of the fundamental result already established in Chapter 2, that an incompressible fluid can be modelled as an added mass matrix noted here ⎡⎣ M f ⎤⎦ operating on the degrees of freedom of the structure solely. Since ⎡⎣ M f ⎤⎦ is positive definite, the fluid lowers the natural frequencies of the structure and if ⎡⎣ M f ⎤⎦ is nondiagonal, the mode shapes are changed with respect to those in vacuum. If the constrained model [6.253] is discretized instead of the spring model [6.237], the corresponding finite element model is of the type:
Vibroacoustic coupling
⎡[ K ] [ 0] ⎢ s ⎢ [ 0] ⎣⎡ K f ⎦⎤ ⎢ T ⎢⎣[ L] − [ L]T
[ L] ⎤⎥ ⎡ [ X s ] ⎤ ⎡[ M s ] ⎢ ⎥ ⎢ − [ L]⎥ ⎢ ⎣⎡ X f ⎦⎤ ⎥ + ⎢ [0] ⎥⎢ ⎥ ⎢ [0] ⎥⎦ ⎣ [ Λ ] ⎦ ⎣ [0]
⎡
[ 0] ⎣⎡ M f ⎦⎤ [ 0]
(e)
⎤
[0]⎤ ⎡⎢ ⎡⎣ X s ⎤⎦ ⎤⎥ ⎢ ⎣⎡ F ⎦⎤ ⎥ ⎥ [0]⎥ ⎢ ⎣⎡ X f ⎦⎤ ⎥ = ⎢⎢ ⎣⎡ S ( e) ⎦⎤ ⎥⎥ ⎢ ⎥ [0]⎥⎦ ⎢⎣ ⎡⎣ Λ⎤⎦ ⎥⎦ ⎢ [0] ⎥ ⎢⎣
575
[6.285]
⎥⎦
When writing down equations [6.281] or [6.285], it is tacitly assumed that the boundary terms of the functionals vanish. Accordingly, they fulfil automatically the conditions of free boundaries, that is pressure is assumed to be zero everywhere on (S f ∩W ) . As in the case of structures, other boundary conditions can be modelled using either a penalty or a Lagrange multiplier method. For instance, a large stiffness coefficient on the diagonal term of the stiffness matrix K f ( j , j ) can be used as a penalty factor to enforce a node of the displacement X f ( j ) . Of course, all of these considerations are replicating those already presented in the case of elastic solids. They are simply recast here in terms of fluid dynamics as a preliminary before extending the method to the mixed formulations. 6.5.3.2 Finite element equations in the X s , p variables
(
)
By discretizing the functionals [6.257] to [6.261] and the volume part of [2.264], the following fluid and vibroacoustic finite element matrices are defined: ⌠ ⎮
Eef′ = ⎮
⎮ ⎮ ⌡(Vf )
pp T dVf [ p ] ⎡⎣ M ′f ⎤⎦ [ p] ce2
⌠
Eif′ = − ⎮⎮
⎮ ⌡(Vf
⌠
Es′f = ⎮⎮
⌡(W ) ⌠
E f′S = ⎮⎮
⎮ ⌡(W )
)
( grad p ) dV 2
= [ p ] ⎡⎣ K ' f ⎤⎦ [ p ] T
f
[6.286]
T pX s .n dW [ X s ] ⎡⎣ K sf′ ⎤⎦ [ p ] T ρ f pX s .n dW [ p ] ⎣⎡ M ' sf ⎦⎤ ⎡⎣ X s ⎤⎦
Furthermore, comparing Esf′ and E fS′ it is noted that the coupling matrices ⎡⎣ K sf′ ⎤⎦ and ⎡⎣ M ' sf ⎤⎦ are very similar and can be suitably written as: ⎡⎣ K sf′ ⎤⎦ = ⎡⎣Csf ⎤⎦ ; ⎡⎣ M sf′ ⎤⎦ = ρ f ⎡⎣Csf ⎤⎦
T
This leads to a finite element model of the generic type:
[6.287]
576
Fluid-structure interaction (e) ⎤ ⎡ ⎡ X s ⎤ ⎤ ⎡ ⎡⎣ F ⎤⎦ ⎤ ⎣ ⎦ ⎥ ⎥⎢ ⎥=⎢ ⎡⎣ M ' f ⎤⎦ ⎥⎦ ⎢⎣ [ p ] ⎥⎦ ⎢ ⎡ S ( e ) ⎤ ⎥ ⎦ ⎦⎥ ⎣⎢ ⎣
⎡[ K s ] ⎣⎡Csf ⎦⎤ ⎤ ⎡[ X ]⎤ ⎡ [ M S ] ⎢ ⎥⎢ s ⎥+⎢ T ⎢ ⎥ [ p ] ⎦ ⎢⎣ ρ f ⎡⎣Csf ⎤⎦ ⎣ [ 0] ⎣⎡ K ' f ⎦⎤ ⎦ ⎣
[ 0]
[6.288]
which is obviously non symmetric. To establish the system of equations [6.288], it is tacitly assumed that the boundary terms of the functionals vanish. As the fluid is described in terms of pressure, it fulfils automatically the dual condition of a free boundary, that is pressure is zero everywhere on the nodes discretizing (S f ∩W ) . It is noted that ⎡⎣ M ' f ⎤⎦ tends to zero as the speed of sound ce tends to infinity. Accordingly, for an incompressible fluid, the modal problem reads as: ⎡ ⎡[ K s ] ⎡Csf ⎤ ⎤ ⎡ [M s ] ⎣ ⎦⎥ ⎢⎢ − ω2 ⎢ T ⎢ ⎢ [ 0] ⎡ K ' f ⎤ ⎥ ⎣ ⎦⎦ ⎣⎢ ρ f ⎣⎡Csf ⎦⎤ ⎣⎣
[0]⎤ ⎤ ⎡[ X s ]⎤ ⎡[0]⎤ ⎥⎥ ⎢ ⎥=⎢ ⎥ [0]⎦⎥ ⎥⎦ ⎣ [ p ] ⎦ ⎣[0]⎦
[6.289]
As in the case of the model [6.281], elimination of pressure gives a modal equation identical to [6.284]. Determination of the added mass matrix [ M a ] is left to the reader as a short exercise. On the other hand, if the pressure is eliminated in the forced problem, one obtains a forced equation of the type:
[ K s ][ X s ] + ⎡⎣[ M s ] + [ M a ]⎤⎦ ⎡⎣ Xs ⎤⎦ = ⎡⎣ F ( e) ⎤⎦ − ⎡⎣Csf ⎤⎦ ⎡⎣ K ' f ⎤⎦
−1
⎡ S (e) ⎤ ⎣ ⎦
[6.290]
Equation [6.290] specifies how the external excitations of the structure and the fluid combine to force the fluid-structure coupled system. Another interesting exercise, borrowed from [SIG 04], is to include in the model the effect of gravity on a water level, denoted ( Σ 0 ) . The point is that a proper choice must be made in order to express the functional of gravity potential energy to fit fruitfully into the X s , p formalism. A natural though unsuited choice is to use
(
)
the formula [1.65], written as: ⌠
Eg = ⎮⎮
⌡( Σ 0 )
ρ f gZ 2 ( x, y , H ; t )dxdy =
where H
1 ρf g
⌠ ⎮ ⎮ ⌡( Σ 0 )
( p( x, y , H ; t ) )
2
dxdy
[6.291]
is the height of the water level. Discretization of Eg produces the
functional: Eg =
Here,
1 ρf g
⌠ ⎮ ⎮ ⌡( Σ 0 )
[ p0 ]
p 2 dxdy = [ p0 ] ⎡⎣ K gf ⎤⎦ [ p0 ] T
is the pressure field restricted to the nodes used to discretize
[6.292]
(Σ 0 ) .
Hence, in order to include the corresponding term into the finite element model [6.288], it is appropriate to split the pressure field into two distinct components [ p0 ]
Vibroacoustic coupling
and
[ p1 ]
where
[ p1 ]
refers to all the remaining components of
577
[ p ] . In order to
alleviate the mathematical manipulations, presentation is restricted here to the problem of sloshing modes. Accordingly, the system [6.288] is transformed into a matrix system of the type: ⎡⎡⎡ K + K ' ⎤ ⎡ K ' ⎤⎤ ⎡ ⎡M ' ⎤ ( 00 ) ⎦ ⎣ ( 01) ⎦ ⎢⎢⎣ g ⎥ − ω 2 ⎢ ⎣ ( 00) ⎦ T T ⎢⎢ ⎢⎡ ⎡ K '(11) ⎤ ⎥⎥ ⎤ M ' ⎢⎣ ⎢⎣ ⎡⎣ K '( 01) ⎤⎦ 01 ( ) ⎢ ⎣ ⎦⎦ ⎦ ⎣⎣
⎡ M '( 01) ⎤ ⎤ ⎤ ⎣ ⎦ ⎥ ⎥ ⎡[ p0 ]⎤ ⎡[ 0]⎤ ⎥=⎢ ⎥ ⎥⎢ ⎡ M '(11) ⎤ ⎥⎥ ⎥ ⎣ [ p1 ] ⎦ ⎣[ 0]⎦ ⎣ ⎦ ⎦⎦
[6.293]
where [ K ′] and [ M ′] are simply the split form of the matrices ⎡⎣ K ' f ⎤⎦ and ⎡⎣ M ' f ⎤⎦ respectively. At this step, it becomes obvious that the choice made to account for the water level energy is unsuited to solve the problem of sloshing modes, since if compressibility of the fluid is discarded the mass matrix of system [6.293] vanishes! In fact, due to the duality between the displacement and the pressure formalism, the suited choice is to formulate the functional of gravity potential energy as a quadratic form involving a mass matrix. Hence, turning to the kinetic energy involved in the vertical motion of the water level, the energy functional may also be written as: Eg′ =
1 ⌠⎮ 2 ( p ( x, y, H ; t ) ) dxdy ⎮ g ⌡( Σ 0 )
[6.294]
Discretization of Eg′ produces the functional: Eg′ =
1⌠ T ⎮ p 2 dxdy = [ p0 ] ⎡⎣ M gf ⎤⎦ [ p0 ] g ⎮⌡( Σ 0 )
[6.295]
Accordingly, equation [6.293] is replaced by the suitable form: ⎡⎡ ⎡K ' ⎤ ⎢ ⎢ ⎣ ( 00) ⎦ T ⎢⎢ ⎢⎣ ⎣⎢ ⎡⎣ K '( 01) ⎤⎦
⎡ ⎡ M gf′ + M ' ⎤ ⎡ M ' ⎤ ⎤ ⎤ ⎡ K '( 01) ⎤ ⎤ ( 00) ⎦ ⎣ ( 01) ⎦ ⎥ ⎡[ p0 ]⎤ ⎡[ 0]⎤ ⎣ ⎦⎥ 2 ⎢⎣ ⎥ −ω = T ⎥ ⎢ ⎥ ⎥ ⎢ [ p1 ] ⎥ ⎢[ 0]⎥ ⎡ K '(11) ⎤ ⎥ ⎡ ⎤ ⎡ ⎤ ⎣ ⎦ ⎣ ⎦ M ' M ' ⎣ ⎦⎦ ⎣ (11) ⎦ ⎦⎥ ⎥⎦ ⎣⎢ ⎣ ( 01) ⎦
[6.296]
As compressibility of the fluid is discarded, [6.296] becomes: ⎡⎡ ⎡K ' ⎤ ⎢ ⎢ ⎣ ( 00) ⎦ T ⎢⎢ ⎢⎣ ⎢⎣ ⎣⎡ K '( 01) ⎦⎤
⎡ ⎤⎤ ⎡ ⎣⎡ M gf′ ⎦⎤ ⎣ K '( 01) ⎦ ⎥ − ω2 ⎢ T ⎡ ⎤⎥ ⎣⎢ [ 0] ⎣ K '(11) ⎦ ⎥⎦
[0]⎤ ⎥⎤ ⎡[ p0 ]⎤ ⎡[0]⎤ ⎥ ⎢ ⎥=⎢ ⎥ [0]⎦⎥ ⎥⎥ ⎣[ p1 ] ⎦ ⎣[0]⎦
[6.297]
⎦
Elimination of [ p1 ] is immediate, producing the modal equation: ⎡ ⎡ K ' ⎤ − ⎡ K ' ⎤ −1 ⎡ K ' ⎤ T − ω 2 ⎡ M ′ ⎤ ⎤ [ p ] = 0 ⎣ gf ⎦ ⎥⎦ 0 ⎢⎣ ⎣ ( 00) ⎦ ⎣ (11) ⎦ ⎣ ( 01) ⎦
[6.298]
The remarkable feature of equation [6.298] is that, in agreement with the physics of the problem, it is expressed in terms of the sole DOFs which are necessary to
578
Fluid-structure interaction
describe the surface waves. Of course, elimination of
[M ′]
[ p1 ]
is no more possible if
differs from zero, that is if the fluid is compressible. This impossibility marks
the presence of coupling between the sloshing and the acoustical modes of the fluid volume. Finally, if a vibrating structure is included in the incompressible model, [ p0 ] and [ X s ] are found to be coupled together, as already highlighted in Chapter 3. 6.5.3.3 Finite element equations in the X s , Π , p variables
(
)
The discretized functionals serving to define the finite element matrices of interest are: Eef′′ =
⌠ ⎮ ⎮ ⎮ ⎮ ⌡(Vf ) ⌠
Esf′′ = ⎮⎮
⌡(W )
⌠
⎮ Π p T dVf [ Π ] ⎡⎣ M Π′′ p ⎤⎦ [ p ] ; Eif′′ = − ⎮⎮ ρ f ce2 ⎮
(
⌡(Vf )
⌠ T Π X s .ndW [ X s ] [ M S′′Π ] ⎡⎣ Π ⎤⎦ ; E fS′′ = ⎮⎮
⌠ ⎮
EiH′′ = − ⎮⎮
1 gradΠ ρf
⌡(W )
⌠
⎮ ⌡(Vf )
⎮ T pΠ ′′ = ⎮⎮ dVf ⎡⎣ Π ⎤⎦ ⎡⎣ M Π′′ p ⎦⎤ [ p ] ; EeH 2 ρ f ce ⎮
⌡(Vf )
)
2
′′ ][ Π ] dVf − [ Π ] [ M ΠΠ T
T Π X s .n dW [ Π ] [ M s′′Π ][ X S ] p2 T dVf [ p ] ⎡⎣ K ′′pp ⎤⎦ [ p ] ρ f ce2
[6.299] This leads to a finite element model of the generic type: ⎡[ K ] ⎢ s ⎢ [ 0] ⎢ ⎢⎣ [ 0]
[ 0] [ 0] [ 0]
⎤ ⎡[ X ]⎤ ⎡ [ M s ] ⎥⎢ s ⎥ ⎢ T ⎥ ⎢ [ Π ] ⎥ + ⎢[ M s′′Π ] ⎢ ⎥ ⎡⎣ K ′′pp ⎤⎦ ⎥⎦ ⎣⎢ [ p ] ⎦⎥ ⎢ 0 ⎣ [ ]
[ 0] [ 0]
[ M s′′Π ] ′′ ] − [ M ΠΠ ⎣⎡ M Π′′ p ⎦⎤
T
⎤ ⎡ ⎡ X ⎤ ⎤ ⎡ ⎡ F ( e ) ⎤ ⎤ ⎦⎥ ⎥ ⎢⎣ s ⎦⎥ ⎢⎣ (e) ⎤ ⎥ ⎢ ⎥ ⎡ ⎢ ⎥ ′′ ⎡⎣ M Π p ⎤⎦ ⎡⎣ Π ⎤⎦ = S ⎦⎥ ⎥⎢ ⎥ ⎢⎣ ⎢ [ 0] ⎥ ⎥ p [ ] ⎢ ⎥ [ 0] ⎦ ⎣ ⎦ ⎣⎢ ⎦⎥
[ 0]
[6.300] If the fluid is incompressible this system implies: −1 ′′ ] ⎡ S ( e ) ⎤ [ K s ][ X s ] + ⎣⎡[ M s ] + [ M a ]⎦⎤ ⎡⎣ Xs ⎤⎦ = ⎡⎣ F ( e) ⎤⎦ − [ M s′′Π ][ M ΠΠ ⎣ ⎦
′′ ] The added mass matrix [ M a ] = [ M s′′Π ] [ M ΠΠ
−1
[6.301]
[ M s′′Π ]T is the same as that produced
by the ( X s , p ) formalism. 6.5.3.4 Example: 1D acoustic finite element As an example, let us consider the one-dimensional case of a tube element of length n delimited by the two nodes I and J. Restricting here the problem to the fluid terms of the system [6.300], a linear interpolation of Π and p between I and J
Vibroacoustic coupling
579
is appropriate, since the differential operator describing the fluid motion is of the second order. Thus, the acoustic interpolation functions are: 0 ⎡ Π n ( x ) ⎤ ⎡ ( Π J − Π I ) x / n + Π I ⎤ ⎡1 − x / n ⎢ ⎥=⎢ ⎥=⎢ 1− x / n ⎣ pn ( x ) ⎦ ⎣ ( pJ − pI ) x / n + pI ⎦ ⎣ 0
x / n 0
⎡Π I ⎤ 0 ⎤ ⎢ pI ⎥ ⎢ ⎥ x / n ⎥⎦ ⎢ Π J ⎥ ⎢ ⎥ ⎣ pJ ⎦
[6.302] Substituting [6.302] into the functionals [6.299] and assuming that the crossn sectional area S (f ) and the Young modulus ρ f ce2 are uniform within the element n, the following element functional and matrices are found:
Ee(f ) = n
Eif( ) = n
(n)
Sf n 6ρ f ce2
(n)
Sf
ρ (f n ) n
[Π I
[Π I
S (f )
pI
pI
ΠJ
1 0 1 / 2 ⎤ ⎡ Π I ⎤ ⎡ 0 ⎢ ⎥ ⎢ 1 0 1 / 2 0 ⎥⎥ ⎢ p I ⎥ pJ ] ⎢ ⎢ 0 1/ 2 0 1 ⎥ ⎢ Π J ⎥ ⎥ ⎢ ⎥⎢ 1 0 ⎦ ⎣ p J ⎦ ⎣1 / 2 0
[6.303]
ΠJ
⎡1 ⎢0 pJ ] ⎢ ⎢ −1 ⎢ ⎣0
0 −1 0⎤ ⎡ Π I ⎤ 0 0 0⎥ ⎢ pI ⎥ ⎥ ⎥⎢ 0 1 0⎥ ⎢ Π J ⎥ ⎥ ⎥⎢ 0 0 0⎦ ⎣ pJ ⎦
[6.304]
n
(n) EeH
=
n ⎡Π 6ρ f ce2 ⎣ I
p
I
Π
J
⎡0 0 ⎢0 1 p ⎤⎢ J ⎦ ⎢0 0 ⎢ ⎣ 0 1/ 2
⎡ Π ⎤ 0 0 ⎤⎢ I ⎥ p ⎥ 0 1/ 2⎥ ⎢ ⎥⎢ I ⎥ 0 0 ⎥ ⎢ Π ⎥ J ⎥ ⎥ 0 1 ⎦ ⎢ ⎢⎣ pJ ⎥⎦
[6.305]
Whence the modal equation: ⎡ ⎢ S (n) ⎢ f n ⎢ 3E ( n ) f ⎢ ⎢⎣
⎡0 0 ⎢0 1 ⎢ ⎢0 0 ⎢ ⎣ 0 1/ 2
−a b / 2⎤ ⎤ ⎡ Π I ⎤ ⎡0⎤ 0 0 ⎤ b ⎡ a ⎥ ⎢ b 0 1/ 2 ⎥⎥ 0 b /2 0 ⎥⎥ ⎥ ⎢⎢ pI ⎥⎥ ⎢⎢ 0 ⎥⎥ n − ω 2 S (f ) ⎢ = ⎢ −a b / 2 a 0 0 ⎥ b ⎥ ⎥ ⎢Π J ⎥ ⎢0⎥ ⎥ ⎢ ⎥ ⎥ ⎢ ⎥⎥ ⎢ b 0 1 ⎦ 0 0 ⎦ ⎥⎦ ⎣ p J ⎦ ⎣ 0 ⎦ ⎣b / 2
[6.306] where a = 1/ ρ (f ) n and b = n / 3E (f ) . As expected, the “stiffness” matrix is n
n
singular and its physical dimension differs from that of a stiffness coefficient, whereas the “mass” matrix includes coefficients of distinct physical dimensions.
580
Fluid-structure interaction
Therefore, it may be preferred to recast the equation by using dimensionless variables based on the scale factors [6.71] already introduced in section 6.2. As a short exercise, the nodal variables Π I and Π J can be eliminated by using the second and the fourth row of [6.306], which merely state that Π I = pI / ω 2 and Π J = pJ / ω 2 . In the process, the system [6.306] is condensed into: ⎡ S (f n ) ⎢ (n) ⎢⎣ ρ f n
S (f n ) n ⎡ 1 −1⎤ 2 − ω ⎢ −1 1 ⎥ n 3E (f ) ⎣ ⎦
⎡ 1 1/ 2⎤ ⎤ ⎡ pI ⎤ ⎡0 ⎤ ⎢1/ 2 1 ⎥ ⎥ ⎢ p ⎥ = ⎢0 ⎥ ⎣ ⎦ ⎥⎦ ⎣ J ⎦ ⎣ ⎦
[6.307]
For comparison, it is recalled that the modal equation for the longitudinal waves of a straight beam (see [AXI 05], Chapter 3) reads as: ⎡ ES( n ) S S( n ) ⎢ ⎣⎢ n
( n) ( n) ⎡ 1 −1⎤ 2 ρS SS n − ω ⎢ −1 1 ⎥ 3 ⎣ ⎦
⎡ 1 1/ 2 ⎤ ⎤ ⎡ X I ⎤ ⎡0 ⎤ ⎢1/ 2 1 ⎥ ⎥ ⎢ X ⎥ = ⎢0 ⎥ ⎣ ⎦ ⎥⎦ ⎣ J ⎦ ⎣ ⎦
[6.308]
X I and X J are the nodal displacements of the beam element in the axial direction. It can be noted that the dual equations [6.307] and [6.308] can be identified with n each other, provided pressure is scaled by E (f ) . Indeed, multiplying the system
[6.307] by ρ (f
n)
and substituting the dimensionless variables ϖ I = pI / E (f
n)
and
(n)
ϖ J = pJ / E f into it, we arrive at:
⎡ S (f n ) E (f n ) ⎢ ⎣⎢ n
n n ρ (f ) S (f ) n ⎡ 1 1/ 2 ⎤ ⎤ ⎡ϖ I ⎤ ⎡0⎤ ⎡ 1 −1⎤ 2 ω − ⎢ −1 1 ⎥ ⎢1/ 2 1 ⎥ ⎥ ⎢ϖ ⎥ = ⎢0⎥ 3 ⎣ ⎦ ⎣ ⎦ ⎦⎥ ⎣ J ⎦ ⎣ ⎦
[6.309]
Chapter 7
Energy dissipation by the fluid
Material presented in this chapter, the last of this volume, concerns fluid damping, which means the ability of a fluid coupled to a structure to extract some amount of mechanical energy in such an irreversible way that energy delivered to the fluid never returns back to the structure, as mechanical energy at least. Major consequences of introducing dissipation in a dynamical model have been already presented and discussed in several of the preceding parts of this book series and the reader is certainly aware of the importance of damping as a mean to avoid unacceptable response levels of structures at resonances, which is a vital issue to many industrial and even common life applications. Therefore we expect that the reader shall be naturally inclined to bear a special attention to the content of the present chapter, at the risk of feeling frustrated to learn here just a few topics on such an important mater. Hence, it is in order to recall once more that the aim of the present book series is limited to equip the reader with a good basic understanding of the physical mechanisms and mathematical methods of general interest, which can serve as a springboard to tackle advanced and specialized topics. Fluid damping can rightly be considered as one among them, which is particularly appealing and challenging. Most of the theoretical and semiempirical informations made available in the open literature on the subject, are presented in accordance with the specific field of application, which may concern typically, architectural and environmental acoustics, naval and ocean engineering, or energy and transport industry. Here presentation is deliberately restricted to a few aspects of the subject which are selected not only for their practical importance but also for relative easiness in mathematical modelling, while mentioning in a very few words several other mechanisms with the sole aim of providing the reader with the references to a few textbooks or specialized papers to initiate its own research work.
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Fluid-structure interaction
7.1. Preliminary survey on linear modelisation of dissipation 7.1.1
Diversity and importance of the dissipative processes
It is recalled that real material systems are always exchanging energy with other systems through various interaction mechanisms. In particular, mechanical systems interact with thermo-dynamical systems and as a general and inevitable result, mechanical energy is finally dissipated into heat. In particular, straining of a continuous medium is never an adiabatic and perfectly reversible process, which reveals the presence of some thermo-mechanical coupling mechanisms in the real behaviour of the material. The amount of mechanical energy which is converted into heat by such processes is irremediably lost by the mechanical system. The physical processes leading to dissipation of mechanical energy are numerous and difficult to model based on sound theoretical background, for most of them at least. However, they can be tentatively classed into the three large following categories: 1. Radiation mechanisms transfer the energy of the vibrating system to waves. As illustrated in Chapter 6 and further analysed in the present chapter, the radiative transfer is irreversible unless the waves are entirely reflected back to the emitting structure. 2. Friction mechanisms convert the mechanical energy into heat. Actually, friction stands for a very broad class of theoretical or semi empirical models, which is as large and diversified as the physical processes involved in friction at the continuous and molecular scale. 3. Non conservative coupling mechanisms are marked by an exchange of mechanical energy between to kinds of motions, namely a permanent motion driven by an external source and a vibration. Depending whether energy is transferred to the permanent motion or to the vibration, the latter is damped or at the opposite amplified. A large part of Volume 4 will be devoted to such mechanisms which are of paramount importance in design engineering against flow induced vibration problems. 7.1.2
The viscous damping model
As already presented in [AXI 04], the most convenient way to account for an irreversible exchange of mechanical energy between two systems in a dynamical analysis, is to assume the presence of interaction forces proportional to the velocity field of the mechanical system. Such a postulate gives rise to the viscous damping model, which allows one to produce analytical solutions using the standard tools of applied mathematics. Although broadly accepted for historical reasons, the name is in fact misleading since it refers to an empirical model adopted for mathematical convenience and not to a physical mechanism of dissipation by viscous friction. This is not to say that such a simplistic model has to be automatically rejected as being unphysical. In fact, the viscous damping model has long been recognized as a very efficient tool to deal with many theoretical and practical problems in dynamics, especially when the rate of dissipation is small. It is nevertheless advisable, and even
Energy dissipation by the fluid
583
necessary, to check its validity by comparing its predictions to those produced by more elaborate models fitting more closely to the physical mechanism of dissipation, as illustrated later in this chapter. The viscous damping model is discussed from the mathematical standpoint, in order to point out a few features of major theoretical importance. With this object in mind, we start from the linear dynamical equations of a mechanical system provided with viscous damping, which read as: e [7.1] K ⎡⎣ X ⎤⎦ + C ⎡⎢ X ⎤⎥ + M ⎡⎢ X ⎤⎥ = F ( ) ( r ; t ) ⎣ ⎦ ⎣ ⎦ Like the stiffness and mass operators, the viscous damping operator C [
],
is
assumed to be real and time independent. To the forced equations [7.1] it is possible to associate a modal equation of the type: ⎡⎣ K + iωC − ω 2 M ⎤⎦ X = 0 [7.2] As in the case of conservative systems, we search for nontrivial solutions of the harmonic type ϕ eiωt . Theory of complex modes is closely related to non self-adjoint operators and is revealed as to be significantly more involved than in case of the self-adjoint and positive operators we dealt with up to here. No attempt will be made to describe here the theoretical aspects of the complex modal theory; the interested reader can be reported to [EVI 00]. For our limited purposes, it is found more appropriate to introduce the subject through a few specific examples which are well suited to grasp the physics which lies behind the formalism. The first important point is that in presence of dissipation, the free oscillations of the system cannot be periodic because mechanical energy is dissipated and no stationary waves can take place. As a consequence, the eigenvalues of the modal problem [7.2] are complex. As already outlined in [AXI 04], the imaginary part characterizes the flow rate of outgoing or ingoing energy, while the real part characterizes the frequency at which the system vibrates. On the other hand, the mode shapes are generally complex. To solidify the understanding of such properties, it is appropriate to review a few typical examples of increasing complexity. The simplest system is the damped harmonic oscillator revisited here for the sake of clarity. 7.1.2.1 Damped harmonic oscillator Considering a harmonic oscillator provided with some viscous damping, the modal problem [7.2] can be conveniently written as: ⎡⎣ω12 + 2iωω1ς 1 − ω 2 ⎤⎦ [ϕ ] = 0 ;
ω12 =
K M
;
ς1 =
C 2ω1 M
[7.3]
Roots of the characteristic equation [7.3] are obviously complex and expressed as:
(
ωc ± = ω1 ± 1 − ς 12 + iς 1
)
[7.4]
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Fluid-structure interaction
The complex frequencies [7.4] imply harmonic motion of complex amplitude: ϕ = ϕ 1e −ω ς t e 1 1
± iω 1t 1−ς 12
[7.5]
where ϕ 1 stands for an arbitrary complex constant. The argument of the first exponential term in [7.5] is real, which implies that the magnitude of the motion decreases or increases exponentially in time, depending on whether the damping ratio is positive or negative. The second exponential with an imaginary argument describes the oscillatory part of the motion, provided the damping ratio remains less than the critical value ς cr = 1 . In terms of real and imaginary parts of the complex pulsation, ς 1 can be expressed as: ς1 =
Im (ωc )
( Re (ω ) ) + ( Im (ω ) ) 2
c
[7.6]
2
c
It is worthwhile stressing that as the two solutions [7.4] are complex conjugate, they stand for the same real vibration which is obtained by taking either the real or the imaginary part of the complex amplitude. Consequently, it suffices to focus only on one of the two solutions [7.4]. By continuity with the convention already adopted in this book for the conservative systems, hereafter we will concentrate on the socalled positive branch ωc + = ω1
(
)
1 − ς 12 + iς 1 . It is also useful to recall that the
decay of energy from one oscillation to the next is described by the dimensionless ratio: ΔE ΔE = 1 − e −2T /τ d ⇒ 4πς 1 if ς 1 1 Em Em
[7.7]
where Em is the mechanical energy, T the pseudo-period of oscillation and τ d = 1/ ω1ς 1 is the damping time. The decrease is thus of about 12% if ς 1 = 0.01 , whereas the magnitude of vibration is lowered by about 6%. On the other hand, if the oscillator is excited by an harmonic force of pulsation ω, a simple calculation shows that the energy dissipated per cycle of vibration can be written as: ΔE = πωCX 2
[7.8]
Relation [7.8] implies that according to the viscous damping model, the amount of energy dissipated per cycle of vibration under harmonic excitation is proportional to the coefficient of viscous damping, to the frequency of the excitation signal and to the square of the vibration amplitude.
Energy dissipation by the fluid
585
Figure 7.1. Width of the peak of the spectral response and damping ratio and Argand plot of the transfer function. Thin line and dots: H = X / F , heavy line and dots: H = zX / F
Finally, it is also useful to relate the width of the peak of the spectral response of the oscillator to the value of the damping ratio. We recall that provided ς 1 is sufficiently small, the squared modulus of the transfer function in displacement is half its maximum value when frequencies are shifted by Δ z = ±ς 1 from the undamped response frequency. This approximate result is illustrated in Figure 7.1, where it is shown than it can be reasonably used in common practice for damping values up to about 10%. The bottom plots of Figure 7.1 are another suitable way to display graphically the damped resonant portion of the transfer function as a so called Argand, or Nyquist plot, in which the imaginary part is plotted versus the real part. It turns out that in the case of viscous damping, an exact circle is obtained, provided the response is the velocity instead of the displacement. Such plots are widely used in modal testing for experimental determination of the parameters of oscillators, or well separated modes of multiple degrees of freedom systems. The reader interested in the subject can be reported in particular to [EWI 00]. 7.1.2.2 Multiple degrees of freedom systems Extension to discrete system with several degrees of freedom leads to natural frequencies and mode shapes which are complex, in most cases at least. The underlying physical reason to obtain a complex mode shape is that due to
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Fluid-structure interaction
dissipation, waves cannot be of the stationary or standing type. In contrast they comprise necessarily a travelling component which conveys the dissipated energy from the source towards the sink of energy. On the other hand, the mathematical form of the dissipative modal problem differs from the conservative one since the modal equation expressed in terms of the Laplace variable s = iω is now of the type: ⎡⎣[ K ] + s [C ] + s 2 [ M ]⎤⎦ [ϕ ] = [0]
[7.9]
Depending on the mathematical properties of the viscous damping [C] matrix, which a priori is not necessarily symmetric, the eigenvalues and the eigenvectors of equation [7.9] can be real, or complex. In any case, since the coefficients of [C] are assumed to be real, it can be stated that the eigenvalues sn of [7.9] occur in complex conjugate pairs, or are real, and so do the eigenvectors. To solve equation [7.9] it is convenient to transform it first into an equivalent form similar to that which holds for a conservative system. It turns out that the appropriate vector transformation is:
[ψ ] = [ϕ
sϕ ]
T
[7.10]
Substituting [7.10] into [7.9], one obtains a matrix equation of the type: ⎡⎣[ A] + s [ B ][ψ ]⎤⎦ = [0]
[7.11]
which leads to the required form [7.10] with the following matrices: ⎡[ K ] [ A] = ⎢ ⎣ [ 0]
[ 0] ⎤ ⎥ − [ M ]⎦
;
⎡ [ C ] [ M ]⎤ [ B] = ⎢ ⎥ ⎣[ M ] [ 0] ⎦
[7.12]
Incidentally, turning back to the time domain, transforming [7.9] into [7.10] is equivalent to replace a set of N differential equations of the second order into a set of 2N differential equations of the first order. The difference with the conservative case is that [ A] is not positive and [ B ] is not necessarily symmetric. In practice the system [7.12] can be solved numerically by using similar techniques as in the conservative case. As an enlightening example of application, we consider the chain of N identical coupled oscillators already used in [AXI 04] as an archetypical system to understand the physics underlying the concepts of modes and travelling waves in the conservative case. Here, viscous damping is introduced into the system by using viscous dampers or dashpots. Furthermore, it turns out to be meaningful to analyse successively the three arrangements depicted in Figure 7.2. Configuration (a) includes a single damper connected between the n-th particle and the ground. In configuration (b) each particle is connected to the ground through an identical damper and finally in configuration (c), the particles are coupled together both by the springs and the dampers.
Energy dissipation by the fluid
587
Figure 7.2. Chain of N identical oscillators provided with viscous dampers
The undamped chain is described by the modal equation: ⎡ 2ω 02 − ω 2 ⎢ 2 ⎢ −ω 0 ⎢ . ⎢ . ⎢ ⎢ 0 ⎢ 0 ⎢⎣
−ω 02 2ω 02 − ω 2 . . 0 0
. . . . . .
. 0 . 0 . . . . 2 . 2ω 0 − ω 2 . −ω 02
⎤ ⎡ X 1 ⎤ ⎡ 0⎤ 0 ⎥⎢ ⎥ ⎢ ⎥ 0 ⎥ ⎢ X 2 ⎥ ⎢ 0⎥ ⎥⎢ . ⎥ ⎢.⎥ . ⎥⎢ ⎥=⎢ ⎥ . ⎥⎢ . ⎥ ⎢.⎥ −ω 02 ⎥ ⎢ X N −1 ⎥ ⎢ 0⎥ ⎥⎢ ⎥ ⎢ ⎥ 2ω 02 − ω 2 ⎥⎦ ⎣⎢ X N ⎦⎥ ⎢⎣ 0⎥⎦
[7.13]
where ω 02 = K / M . Solution of equation [7.13] was already discussed in depth in [AXI 04], Chapter 6. Here it is sufficient to illustrate the results concerning the natural frequencies and the modes viewed as standing harmonic waves for comparison with the damped cases. Figure 7.3 is a plot of the natural frequencies versus the modal order n for a chain comprising N = 40 identical particles M = 10 kg, K = (10π ) /m, 2
which are found to be nearly in harmonic sequence as long as n is sufficiently small with respect to N. Figure 6.4 shows the relative displacement of the particles at nine successive times equally spaced on half a period of the oscillation according to the first mode. They represent nine snapshots of the time-space oscillation: ⎛ ⎛ nπ ⎞ ⎞ X ( n; t ) = Im ([ϕ1 ] eiω1t ) = ⎜ sin ⎜ ⎟ ⎟ sin 2πτ ⎝ ⎝ N +1⎠⎠
[7.14]
where 2πτ = ω1t , ω1 being the natural pulsation of the first mode.
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Fluid-structure interaction
The important point is that in such a standing wave, time and space are separated and the mode shape [ϕ ] is real. Then, at any given time, all the particles vibrate either in phase, or in phase opposition (out of phase) with each others and the snapshots taken at any time τ 1 are the same as those taken at τ 2 = π − τ 1 .
Figure 7.3. Natural frequencies of the undamped chain versus modal order
Figure 7.4. Displacement of the particles at several times during half a period of an oscillation according to the first mode shape. Circles stand for the material points
Energy dissipation by the fluid
589
In configuration (a) the damped chain is described by the modal equation: ⎡ 2ω02 − ω 2 ⎢ 2 ⎢ −ω0 ⎢ . ⎢ 0 ⎢ ⎢ . ⎢ 0 ⎣⎢
−ω02 2ω02 − ω 2 . 0 . 0
. . . . . . . 2ω02 + 2iωω0ς 0 − ω 2 . .
−ω02 .
.
−ω02
.
0 0 .
⎤ ⎡ X 1 ⎤ ⎡0⎤ ⎥⎢ ⎥ ⎢ ⎥ ⎥ ⎢ . ⎥ ⎢0⎥ ⎥⎢ . ⎥ ⎢.⎥ ⎥ ⎢ ⎥ = ⎢ ⎥ [7.15] ⎥ ⎢ Xn ⎥ ⎢. ⎥ ⎥ ⎢ . ⎥ ⎢0⎥ ⎥⎢ ⎥ ⎢ ⎥ 2ω02 − ω 2 ⎦⎥ ⎣⎢ X N ⎦⎥ ⎣⎢0⎦⎥ 0 0 . 0 .
Provided damping is not too large, the natural frequencies are not significantly changed and modal damping ratios ς n highly varies depending on the frequency and the modal displacement at the location of the dashpot. This point is illustrated in Figure 7.5 which refers to a damper located at the tenth particle and provided with such a damping coefficient that the damping ratio for a single mass-spring system would be ς 0 = 20% . The mode shapes are also complex, as illustrated in Figure 7.6. As a consequence, the phase of the vibration differs from one particle to the next, and the snapshots of real motion largely differ from the conservative case, as it can be easily checked by comparing Figures 7.4 and 7.6 where the characteristic features of a standing wave are obviously destroyed.
Figure 7.5. Modal damping ratios for a chain of 40 particles provided with a dashpot ς 0 = 20% at the tenth particle
The dashpot configurations (b) and (c) are also of particular interest because they pertain to the special class of the so-called proportional damping already introduced in [AXI 04], Chapter 7, which corresponds to complex natural frequencies and the same real mode shapes as in absence of dissipation.
590
Fluid-structure interaction
Figure 7.6. Complex mode shape in terms of modulus and phase angle in degrees
Figure 7.7. Displacement of the particles at several times during half a period of an oscillation according to the first mode shape. Full lines refer to the times between 0 and T/4 and dashed lines to times between T/4 and T/2
Energy dissipation by the fluid
591
Figure 7.8. Modal damping ratios for a chain of 40 particles provided with grounded daspots
ς 1 = 20%
Figure 7.9. Displacement of the particles at several times during half a period of an oscillation according to the first mode shape. Full lines refer to the times between 0 and T/4 and dashed lines to times between T/4 and T/2
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Fluid-structure interaction
Figure 7.10. Modal damping ratios for a chain of 40 particles coupled by dashpots
ς 40 = 20%
It is recalled that proportional damping is defined as:
[C ] = α [ M ] + β [ K ]
[7.16]
where K and M stand for the stiffness and mass matrices of the undamped system while α and β are two constants which can be parametrically varied to adjust the values of the damping ratios ς n of any mode 1 ≤ n ≤ N within a prescribed frequency interval [ f1 , f N ] , in such a way that ς n is less than the highest
of the prescribed values ς 1 and ς N , remaining however reasonably large. That a proportional damping matrix does not affect the mode shapes of the conservative system is a mere consequence of the orthogonality of the mode shapes with respect to the stiffness and mass matrices. Modal projection method applied to the damped system leads to the damping ratios: ⎞ 1⎛ α ςn = ⎜ + βω n ⎟ 2 ⎝ ωn ⎠
[7.17]
Suitable values of α and β versus the prescribed damping values ς 1 and ς N are found to be: α=
2ω1ω2 (ς 1ω2 − ς 2ω1 ) 2 (ς 2ω2 − ς 1ω1 ) ; β= ω22 − ω12 ω22 − ω12
[7.18]
Energy dissipation by the fluid
593
Figure 7.11. Modal damping ratios for a chain of 40 particles: grounded and coupling dashpots ς 1 = ς 40 = 2%
Configuration (b) of the grounded dashpots corresponds to a damping matrix proportional to the mass matrix. As shown in Figure 7.8, the damping ratios are a decreasing function of frequency, in agreement with formula [7.17]. On the other hand, Figure 7.9 illustrates the fact that though the mode shape is real as in the undamped case, the mode of vibration does not represent a standing wave because of dissipation. Accordingly, the particles vibrate either in phase, or in phase opposition with each others as in the conservative case, but the snapshots taken at any time τ 1 (full lines in Figure 7.9) are not the same as those taken at τ 2 = π − τ 1 (dashed lines), precisely because the amplitude of the vibration decreases exponentially with time. Configuration (c) of the coupling dashpots corresponds to a damping matrix proportional to the stiffness matrix. As shown in Figure 7.10, the damping ratios are an increasing function of frequency, again in agreement with formula [7.17]. The vibration is of the same type as in configuration (b). Finally, Figure 7.11 illustrates the variation of the damping ratios when both configurations are mixed together in such a way that ς 1 = ς N . Another important point concerning the generic features of the viscous damping matrices is their symmetry. As in each of the above examples [C ] was found to be symmetric, it is of interest to investigate to which physical mechanism could correspond an asymmetric matrix. At this respect, it is recalled first that an asymmetric matrix can always been viewed as resulting from the sum of symmetrical matrix and a skew symmetrical one:
594
Fluid-structure interaction
⎡ A C⎤ ⎡ A γ ⎤ ⎡ 0 δ ⎤ ⎢ D B ⎥ = ⎢ γ B ⎥ + ⎢ −δ 0 ⎥ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ C+D C−D γ = ; δ= 2 2
[7.19]
On the other hand, skew symmetric matrices of the “viscous type” were already identified as gyroscopic coupling matrices in [AXI 04], Chapter 6. It is recalled that for a discrete system such as a mass-point connected to a rotating wheel through linear springs, see Figure 7.12, the gyroscopic matrix, which accounts for the Coriolis forces exerted on the particle, reads as: ⎡ 0 ⎡⎣G ( Ω )⎤⎦ = 2iω ⎢ ⎣− M Ω
+M Ω ⎤ 0 ⎥⎦
[7.20]
Modal analysis of the coupled system is elementary. It is recalled here, that two whirling modes are found, namely a forward mode rotating in the same direction as the permanent rotation at angular speed Ω of the wheel and a backward mode rotating in the opposite direction. Natural frequencies of these modes are real, which means that gyroscopic coupling is a conservative process. On the other hand, the mode shapes are [1 i ] and [1 −i ] , which describe a circular motion of the mass in the backward and the forward direction, respectively. Such whirling systems including corotating fluids will be described in some depth in Volume 4. Here, the specific object to invoke them is to demonstrate that the mechanism of viscous damping corresponds necessarily to symmetric matrices.
Figure 7.12. Material particle connected to a rotating wheel through linear springs
Finally, before leaving the subject of discrete systems, it may be noted that two fluid and even two fluid-structure counterparts to the chain of mass-spring systems can be proposed as a discretized version of continuous systems. As an example, the system depicted in Figure 7.13 is made of a series of large enclosures of volume
Energy dissipation by the fluid
(VE )
595
limited by a free surface of area (SE ) . Connection of one to the next is
achieved by small tubes of length LT and volume (VT ) << (VE ) . The coupled structure consists in a mass-spring system located at the free surface. The system can be used to model fluid-structure interaction involving either the sloshing modes, or the Helmholtz resonances, or even both of them. Dissipation can be modelled as viscous dampers connected to the springs, or as a viscous damping coefficient distributed within the fluid columns, as further discussed in the next subsection.
Figure 7.13. Chain of large enclosures connected by small tubes
7.1.2.3 Damped acoustical modes in a tube Let us consider first a tube of length L and constant cross-section S f , filled with compressible and dissipative fluid and let us assume that dissipation can be modelled according to the viscous model, which means a force per unit length proportional to particle velocity, or in an equivalent way to volume velocity. The local equations of continuity and momentum are therefore written as: S f ∂p ∂q +ρf =0 2 ∂x ce ∂t ∂q ∂p +Cf q + Sf =0 ρf ∂t ∂x
[7.21]
Elimination of volume velocity is immediate, producing the damped wave equation: − ρ f ce2
∂2 p ∂p ∂2 p + C + ρ =0 f f ∂x 2 ∂t ∂t 2
[7.22]
Since both C f and ρ f are assumed to be uniform, proportional damping occurs and the mode shapes are expected to be the same as in the conservative case, provided of course that the terminal impedances are conservative in nature. To prove this, the most expedient way is to have recourse to the modal expansion method, using the mode shapes of the conservative system as a vector basis to expand the solution of the dissipative modal problem. By virtue of orthogonality of these vectors with
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Fluid-structure interaction
respect to the “mass”, the “stiffness” and damping operators, the series reduce to a single term, which is: ⎛ ⎛ c ( nπ − (α + α ) ) ⎞ 2 ⎞ Cf e in out − ω2 ⎟ = 0 an ⎜ ⎜ ⎟ + iω ⎟ ⎜ ⎜⎝ ⎟ ρf L ⎠ ⎝ ⎠
[7.23]
By comparing [7.23] to [7.3], the following modal damping ratio is found: ςn =
Cf L
2 ρ f ce ( nπ − (αin + αout ) )
[7.24]
The natural frequency and mode shape [4. 4] are repeated here for convenience: fn = ( p)
ϕn
ce ( nπ − (αin + αout ) ) 1 − ς n2 2π L
⎛ x ( nπ − (αin + αout ) ) ⎞ ( x ) = sin ⎜ + αin ⎟ ⎜ ⎟ L ⎝ ⎠
[7.25]
In full agreement with what was said in the context of discrete systems, in presence of proportional damping the acoustical modes stand for damped waves where time and space are separated. The mode shape is that of the standing wave [7.25] and the vibration decays in time as an exponential controlled by the modal damping time: τ d( ) = 1/ ωc( )ς n n
n
[7.26]
where here ωc( ) designates the natural pulsation of the n-th mode of the undamped system. Let us consider now the case of a damper concentrated at some place along the tube, for instance at the outlet. In the absence of the damper the outlet impedance would be conservative: n
Zout =
p q
[7.27] x=L
In the presence of the damper, the momentum equation [7.21] reads as: ρf
∂q ∂p + C f Lqδ ( L − x ) + S f =0 ∂t ∂x
[7.28]
Integrating [7.28] in the interval L − ε , L + ε and letting ε tend to zero, contribution of the first term in [7.28] is found to vanish since in the absence of external source of fluid material q must be continuous. Contribution of the last term comprises the two pressure terms:
Energy dissipation by the fluid
p( L − ε ) = p ( L ) lim ε → 0
;
597
p( L + ε ) = Zout q ( L ) lim ε → 0
Accordingly, the following relation is obtained: q ( L ) ( C f L + Zout ) − S f p ( L ) = 0 ⇒
C L p ( L) = Zout + f q ( L) Sf
[7.29]
Relation [7.29] can be interpreted as new impedance, which takes the damper into account. For instance, if the original impedance was a pressure node, the modified impedance is C f L / S f , if it was a volume velocity node, the impedance remains unchanged, as it should since the damper has no effect. Shifting to the spectral domain, in terms of complex amplitudes, the corresponding dimensionless impedance, denoted γ out , is obtained by solving the transcendental equation: tan γ out = tan α out + i
Cf L
[7.30]
ρ f ce
For the sake of simplicity hereafter, tan αout is assumed to be zero. Then relation [7.30] simplifies into: tan γ out = −i
Cf L ρ f ce
⇔ tanh γ out =
Cf L ρ f ce
=−
Cf L
Z(
sp )
[7.31]
Relation [7.31] highlights the existence of a critical value for the damping coefficient C f L above which no wave is reflected back. Its value is precisely equal to the specific impedance of the fluid already introduced in Chapter 4, subsection 4.2.1.2. Such a result must not surprise us as we already shown in subsections 4.2.1.3 and subsections 4.2.1.6 that for such specific impedance no wave is reflected. Restricting the presentation to the range of small damping, the dissipative and dimensionless outlet impedance is: tan γ out γ out = iC f L / Z (
sp )
[7.32]
Substituting the inlet and outlet impedances into the general formulas [7.25], the following complex natural frequencies and mode shapes are arrived at: fn =
C nce +i f 2L 2πρ f
⎛ nπ x xC f ⎞ p +i ; ϕ n( ) ( x ) = an sin ⎜ ⎟ ⎜ L ρ f ce ⎟⎠ ⎝
where an is an arbitrary real, or complex constant. Using relation [7.6], the modal damping ratio is found to be:
[7.33]
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Fluid-structure interaction
ςn =
Cf L ⎛ Cf L ρ f ce nπ 1 + ⎜ ⎜ ρ c nπ ⎝ f e
⎞ ⎟⎟ ⎠
2
Cf L ρ f ce nπ
[7.34]
Figure 7.14. Snapshots of the pressure field corresponding to the first acoustic mode, for a tube open at the inlet and terminated at the outlet by a viscous damper. Dimensionless times τ are scaled by the natural period of the undamped mode
The mode shape [7.33] can be further transformed by separating the real and imaginary parts and expressed in terms of the natural pulsations and modal damping ratios as: ⎛ ⎛ nπ x ⎞ ⎛ xC f ⎞ ⎛ xC f ⎞ ⎞ ⎛ nπ x ⎞ ϕ n( p ) ( x ) = an ⎜ sin ⎜ + i cos ⎜ cosh ⎜ sinh ⎜ ⎟ ⎟ ⎟ ⎟ ⎜ρ c ⎟ ⎜ ρ c ⎟⎟ ⎟ ⎜ ⎝ L ⎠ ⎝ L ⎠ ⎝ f e⎠ ⎝ f e ⎠⎠ ⎝ ( p)
ϕn
⎛ ⎛ω x ⎞ ⎛ω ς x ⎞ ⎛ω x ⎞ ⎛ ω ς x ⎞⎞ ( x ) = an ⎜ sin ⎜ n ⎟ cosh ⎜ n n ⎟ + i cos ⎜ n ⎟ sinh ⎜ n n ⎟ ⎟ ⎜ ⎟ ⎝ ce ⎠ ⎝ ce ⎠ ⎝ ce ⎠ ⎠ ⎝ ⎝ ce ⎠
[7.35]
The phase of the modal pressure field varies continuously along the tube as: ⎛ω ς x ⎞ ⎛ ω x ⎞ ⎪⎫ ⎪⎧ Ψ n ( x ) = tan −1 ⎨ tanh ⎜ n n ⎟ tan ⎜ n ⎟ ⎬ ⎪⎩ ⎝ ce ⎠ ⎝ ce ⎠ ⎪⎭
[7.36]
On the other hand, the complex amplitude of the harmonic vibration corresponds to a real motion given either by the real or the imaginary part of the complex amplitude, indiscriminately. Selecting the imaginary part, for instance, the following vibration is obtained, where the space and time variables are not separated:
Energy dissipation by the fluid
599
Im (ϕ n ( x )eiωn t ) = ⎛ ω nς n x ⎞ ⎛ ω n x ⎞ ⎛ ω nς n x ⎞ ⎛ ωn x ⎞ ⎪⎧ ⎪⎫ ⎨cosh ⎜ ⎟ sin ⎜ ⎟ sin ωn t + sinh ⎜ ⎟ cos ⎜ ⎟ cos ωn t ⎬ ⎪⎩ ⎪⎭ ⎝ ce ⎠ ⎝ ce ⎠ ⎝ ce ⎠ ⎝ ce ⎠
[7.37]
Hence, as in the case of discrete systems, if damping is not proportional, the snapshots of real motion largely differ from the conservative case, see Figure 7.14: In particular, it can be noted that pressure differs significantly from zero at the outlet, even if the real part of the dimensionless impedance corresponds to a pressure node. Indeed, this is a mere consequence of the fact that, in contrast with the conservative case, the terminal impedance corresponding to a dissipative pressure node is not zero. The result can also be interpreted in terms of energy dissipation, as set mathematically later in subsection 7.2.1.2. Here suffice to say that if pressure would vanish at the outlet, no acoustical energy could be dissipated there, which would mean that the dashpot would have no effect, in clear contradiction to reality. On the other hand, during a cycle of vibration, the pressure antinode is steadily drifting from the left, to the right, that is from the conservative inlet impedance towards the outlet dissipative impedance, which would suggest that the fluid oscillation behaves as some kind of travelling mode, in contrast with the standing wave which occurs in the conservative case. Actually, such a feature can be understood recalling that because the tube outlet stands for a sink of acoustical energy, there is necessarily a flux of energy directed from the tube inlet toward the tube outlet. 7.1.2.4 Transfer matrix method It is of interest to include viscous damping in the transfer matrix method, which was introduced in Chapter 4 in the conservative case. Let us consider a uniform tube element described by the transfer matrix [4.113] repeated here for convenience: ⎡ ωx cos ⎢ ce A ( x; ω ) = ⎢ ⎢ i ρ f ce ωx sin ⎢− S ce f ⎣⎢
−
iS f
ωx ⎤ i ⎤ ce ⎥⎥ ⎡ cos kx − sin kx ⎥ =⎢ Z ⎥ ⎢ ⎥ ωx cos kx ⎦ cos ⎥ ⎣ −iZ sin kx ce ⎦⎥
ρ f ce
sin
[7.38]
Once more ce is the equivalent phase speed of sound in the tube and k = ω / ce is the wave number. As indicated by the modal calculation made in the last subsection, proportional damping corresponds to complex natural frequencies and real mode shapes. The problem is revisited here from the TMM standpoint. Assuming for instance the case of a tube terminated by two conservative pressure nodes, the modal problem reads as: i ⎡ ⎤ − sin kL ⎥ ⎡ q1 ⎤ ⎡ q2 ⎤ ⎢ cos kL = Z ⎢0⎥ ⎢ ⎥⎢0⎥ ⎣ ⎦ −iZ sin kL cos kL ⎦ ⎣ ⎦ ⎣
[7.39]
600
Fluid-structure interaction
As in the conservative case, the homogeneous equation corresponding to the second row of the system [7.38] leads necessarily to the real wave numbers and mode shapes: kn =
nπ L
⎛ nπ x ⎞ p ; ϕ n( ) ( x ) = an sin ⎜ ⎟ ⎝ L ⎠
[7.40]
In presence of proportional damping, the only way to make the result [7.40] compatible with the fact that the natural frequency is complex, is to assume that the phase speed of sound itself becomes complex: c nπ ωn( ) nπ ce kn = = (c) = L ce
(
1 − ς n2 + iς n (c)
)⇒c
(c)
e
Lce
= ce
(
1 − ς n2 + iς n
)
[7.41]
Actually, the concept of complex phase velocity is not so unnatural considering that phase velocity governs precisely the phasing of the wave along the tube which is affected by dissipation as already described in particular by the relation [7.36]. Furthermore, by substituting the result [7.24] into [7.41], the phase speed can be expressed directly in terms of the properties of the dissipative fluid contained in the tube: (c)
ce = ce
(
2 n
1 − ς + iς n
)
⎛ ⎛ C L = ce ⎜ 1 − ⎜ f ⎜ 2 ρ nπ ⎜⎜ ⎝ f ⎝
2
⎞ Cf L ⎟⎟ + i 2 ρ f nπ ⎠
⎞ ⎟ ⎟⎟ ⎠
[7.42]
Formula [7.42] shows that in presence of proportional damping, not only the phase speed becomes complex but also it varies with n, which indicates that the damped waves are also dispersive. Another way to arrive to the same results is to build directly the pertinent transfer matrix related to the damped wave equation [7.22]. The starting point is the harmonic, or Fourier transform of it, which reads as: d2p ⎛ ω ⎞ +⎜ ⎟ dx 2 ⎝ ce ⎠
2
⎛ Cf ⎞ ⎜⎜ 1 − i ⎟p=0 ρ f ω ⎟⎠ ⎝
[7.43]
Once more the general solution is of the type: p ( x; ω ) = A+ eλ+ x + A− eλ− x q ( x; ω ) =
iS f ρfω
(A λ e +
+
λ+ x
; λ± = ∓i + A− λ− e
λ− x
ωη ce
; η = 1−
iC f ρfω
[7.44]
)
Identification of the integration constants to the volume velocity and pressure at the inlet leads to the damped transfer matrix equation:
Energy dissipation by the fluid
iη ⎡ ⎤ − sin kη x ⎥ ⎢ cos kη x Z ⎥ A ( x; ω ) = ⎢ ⎢ −iZ sin kη x cos kη x ⎥ ⎢⎣ η ⎥⎦
601
[7.45]
As a short application, the matrix [7.45] can be applied to determine the acoustic modes of a tube limited by two conservative pressures nodes. Modal equation reads as: iη ⎡ ⎤ − sin kη L ⎥ cos kη L ⎡ q2 ⎤ ⎢ ⎡q ⎤ Z ⎥ ⎢ 1⎥ ⎢ 0 ⎥ = ⎢ −iZ 0 ⎣ ⎦ ⎢ sin kη L cos kη L ⎥ ⎣ ⎦ ⎣⎢ η ⎦⎥
[7.46]
To verify equation [7.46], the modal pressure field is necessarily proportional to sin kη x where the suitable values for kη are the roots of the dispersion equation, which is complex: sin kη L = 0 ⇔ kη =
nπ L
[7.47]
ω . Then denoting k = a + ib , a and b ce are solutions of the two following algebraic equations:
Since η is a complex quantity, so does k =
2 bC f ⎛ nπ ⎞ 2 2 ⎜ ⎟ = a −b + ρ f ce ⎝ L ⎠
[7.48]
⎛ C ⎞ 0 = a ⎜ 2b − f ⎟ ⎜ ρ f ce ⎟⎠ ⎝
Solving [7.48] is immediate, giving the solution: 2 ⎛ nπ ⎞ ⎛ C f ⎞ an = ⎜ ⎟ ⎟ −⎜ ⎝ L ⎠ ⎜⎝ ρ f ce ⎟⎠
2
; bn =
Cf
[7.49]
2 ρ f ce
With the aid of [7.49], the complex natural frequencies are easily found and identified to those already derived by the modal expansion method, (relations [7.24] and [7.25]). The result is written here in the canonical form which holds for proportional viscous damping f n(
c)
= fn
(
)
1 − ς n2 + iς n , where in the present
example: fn =
nce 2L
; ςn =
Cf 2nπρ f ce
[7.50]
602
Fluid-structure interaction
To include proportional viscous damping in the transfer matrix method a particularly simple manner is to replace the real frequencies by the complex values taking into account dissipation. If damping is sufficiently small it suffices to change f into f (1 + iς ) where ζ is entered as an adjustable input. On the other hand, it is also of theoretical and practical importance to analyse the case where dissipation is essentially concentrated at the terminal impedances, in such a way that fluid itself can be considered as conservative. Actually, carrying out the modal calculation in such a case by using the TMM method is very similar to that already presented in subsection 7.1.2.3. Assuming as a typical example, a conservative pressure node at the inlet and a dissipative pressure node at the outlet, the modal equation is therefore written as: ⎡ q2 ⎤ ⎡ i ⎤ − sin kL ⎥ ⎡ q1 ⎤ ⎢ C L ⎥ ⎢ cos kL = Z ⎢− f q ⎥ ⎢ ⎥⎢0⎥ cos kL ⎦ ⎣ ⎦ ⎢⎣ S f 2 ⎥⎦ ⎣ −iZ sin kL
[7.51]
Eliminating q2 between the two equations [7.51] gives the dispersion equation: Cf L ρ f ce
= i tan kL
[7.52]
As fluid is assumed here to be conservative, ce must be a real quantity. Therefore the solutions of equation [7.52] are in terms of complex wave numbers and frequencies, in full agreement with the conclusions of subsection 7.1.2.3. It is convenient to transform equation [7.52] by using the following trigonometric identity: tan( a + ib) =
sin a cosh b + i cos a sinh b cos a cosh b − i sin a sinh b
[7.53]
Then the solutions of [7.52] are of the complex type: ⎛C L⎞ ⎛ C L ⎞⎞ c ⎛ c kn L = nπ + i tanh −1 ⎜ f ⎟ ⇔ ωn( ) = e ⎜ nπ + i tanh −1 ⎜ f ⎟ ⎟ ⎜ρ c ⎟ ⎜ ρ c ⎟⎟ L ⎜⎝ ⎝ f e⎠ ⎝ f e ⎠⎠
[7.54]
It is noted that the analytical form [7.54] differs from that which holds in the case of proportional damping, since here the real parts of the complex natural frequencies are found to be independent from the damping coefficient. The practical importance of such a difference is further illustrated in the next subsection, see in particular Figure 7.15. Recalling that complex terminal impedance implies some amount of reflected and transmitted waves at the tube end, it is also of interest to revisit the concept of specific impedance in the context of the transfer matrix method. As already discussed in Chapter 4, subsection 4.2.1.6, the specific impedance corresponds to a condition of no reflection and full transmission of the incident wave. As a
Energy dissipation by the fluid
603
T
mathematical consequence, the acoustic vector ⎡⎣ q ( x; ω ) p ( x; ω )⎦⎤ must be independent of x, which stated otherwise means that it must be an eigenvector of the transfer matrix. The eigenvalue problem reads as: i ⎡ ⎤ ⎢ cos kL − λ − Z sin kL ⎥ ⎡ q(0; ω ) ⎤ = ⎡0⎤ ⎢ ⎥ ⎢⎣ p(0; ω ) ⎥⎦ ⎢⎣0⎥⎦ ⎣ −iZ sin kL cos kL − λ ⎦
[7.55]
The solution is straightforward. The characteristic equation is found to be: ⎧ λ = e − ikx λ 2 − 2λ cos kx + 1 = 0 ⇒ ⎨ 1 + ikx ⎩λ2 = e
[7.56]
The corresponding eigenvectors normalized to a unit pressure follow as: ⎡ S [ψ 1 ] = ⎢ + f ⎣⎢ ρ f ce
T
⎤ 1⎥ ; ⎦⎥
⎡ S [ψ 2 ] = ⎢ − f ⎣⎢ ρ f ce
⎤ 1⎥ ⎦⎥
T
[Ψ 1 ]
stands for a forward wave associated to the local impedance:
Z+ =
ρ c ρ c Ψ 1 ( 2) = + f e = −i f e tan α + ⇒ tan α + = +i Ψ 1 (1) Sf Sf
[Ψ 2 ] Z− =
[7.57]
[7.58]
stands for a backward wave associated to the local impedance: ρ c ρ c Ψ 1 ( 2) = − f e = +i f e tan α − ⇒ tan α − = −i Ψ 1 (1) Sf Sf
[7.59]
In agreement with the identity [7.53], the dimensionless impedances follow as: α + = +i ∞
;
α − = −i ∞
[7.60]
Such specific impedances can also be rightly termed iterative impedances since they correspond to a condition of wave propagation such that at any place along the tube, the wave can be iterated from the source wave simply by multiplying the latter by the corresponding eigenvalue [7.56] which accounts for the propagation delay. Of course results of the present analysis fully agree with those derived in Chapter 4, subsection 4.2.1.6, based on a maybe less abstract approach, which consisted in identifying directly the forward and backward travelling harmonic waves which fit to the boundary conditions. This point of view can be shortly summarized here as follows. The general system of harmonic plane wave in the tube is described by the vector
604
Fluid-structure interaction
⎡ Sf ⎤ A− eik x − A+ e − ik x ) ⎥ ⎡ q ( x; ω ) ⎤ ⎢ − ( ⎢ ⎥ = ⎢ ρ f ce ⎥ ⎣ p ( x; ω ) ⎦ ⎢ ⎥⎦ A− eik x + A+ e − ik x ⎣
[7.61]
If the tube outlet is terminated by iterative impedance, the following condition must be fulfilled: ⎛ ρ c ⎞ ⎛ A eik L + A+ e − ik L ⎞ ρ f ce p ( L; ω ) = − ⎜ f e ⎟ ⎜ − ik L = ⎜ S ⎟ A e − A e − ik L ⎟ q ( x; ω ) Sf ⎠ + ⎝ f ⎠⎝ −
[7.62]
Condition [7.62] implies necessarily that A− = 0 or, in other terms, no wave is reflected back at the tube end. Accordingly, the iterative impedance can also be rightly termed anechoic impedance which is the most appropriate term to fit closely to the physical reality. 7.1.3
Forced damped waves
7.1.3.1 Spectral domain As stated in the last subsection, the transfer matrix method can be used as a suitable tool to include the effect of proportional and concentrated damping into the spectral analysis of the acoustic response of piping systems. As an example of application let us consider a tube of length L = 25 m and internal radius R = 10 cm, filled with water ( ce = 1000 m/s , ρ f = 1000 kgm -3 ). At the inlet, there is a conservative pressure node and a source of pressure characterized by a flat spectrum of 1Pa 2 Hz -1 in the frequency range of interest.
Figure 7.15. Resonance peak: fit of the proportional damping model to the 1 DOF model
Energy dissipation by the fluid
605
Figure 7.16. Resonance peak: dissipative terminal impedance
Validity of the proportional damping model is checked first by assuming that the tube inlet and outlet are terminated by conservative pressure nodes. Figure 7.15 displays two response peaks which correspond to the first excited resonance as measured at midspan of the tube and two distinct values of the damping ratio ς 1 . The data computed by using the transfer matrix method (dotted lines) and those computed by using the equivalent damped harmonic oscillator (dashed lines) are plotted within the frequency interval (1 ± ς 1 ) f1 . Finally the horizontal dot dashed line marks the half power density level of the spectrum, as in Figure 7.1. The fit between the TMM model and the single DOF model is found to be satisfactory, especially in the low damping domain less than about 0.2. For larger damping ratios, significant discrepancies are noticeable in the wings of the resonance peak, because of the contribution of the other resonances to the response of the TMM model. In a similar manner, validity of the damping model for a terminal impedance is checked by setting proportional damping to zero and assuming that the tube outlet is terminated by a dissipative pressure node. Therefore, the outlet dimensionless impedance is a purely imaginary number varied from nearly zero up to nearly one to cover the nearly conservative case up to the anechoic case. Figure 7.16 displays the response peaks produced by the TMM and the single DOF models. Again, both models give essentially the same results in the low damping range (damping ratios less than about 5%) but differ very importantly when damping is large, especially
606
Fluid-structure interaction
near the anechoic condition which can be accounted for rather suitably by the transfer matrix method, though accuracy is likely spoiled to some extent due to nearly singular and badly scaled matrix. This point is illustrated by the last plot of Figure 7.16, which shows that a residual resonance peak is still computed even if tanh α out is very close to unity. Of course, for numerical reasons the exact anechoic condition tanh αout = 1 cannot be achieved, as it leads unavoidably to a “not a number” diagnostic in MATLAB.
Figure 7.17. Amplitude level of the resonance peak versus terminal damping
On the other hand, it is also of interest to investigate numerically the effect of dissipation at the tube end on the magnitude of the resonant response to a given acoustical source. Taking the same example as just above, we consider the case of a harmonic pressure source of unit amplitude p0 = 1 Pa which is in resonance with the first acoustical mode of the tube f1 = 20 Hz . Figure 7.17 is a plot using logarithmic scales of the relative magnitude of the pressure response at tube midspan versus the damping ratio, which illustrates the importance of limiting dissipation at the tube ends to obtain high levels of resonant responses inside the tube, as is suitable in musical instruments for instance. The fact that the curve does not tend exactly to the expected value of unity as the termination becomes anechoic ( ς 1 → 1 ) is tentatively attributed to numerical errors related to bad conditioning of the transfer matrix. To conclude this subsection, it must be recalled that modal expansion method is also available to study the forced responses in the spectral domain. A practical advantage to express the response as a modal series, is the convenience to adjust suitably the modal ratio of each resonance peak at will, based typically on experimental data, or design rules. However, there is no much to add here on the subject which has been already illustrated in several applications in this book. Let us
Energy dissipation by the fluid
607
emphasize once more that validity of the method is essentially restricted to the range of low damping corresponding to a negligible coupling of the modes by damping. This difficult aspect of the problem is further discussed in the next subsection in the context of analyses performed in the time domain. 7.1.3.2 Time domain: dissipative terminal impedance Wave propagation in a uniform tube provided with a dissipative termination can be calculated analytically without difficulty, provided dissipation within the tube itself can be neglected. As a first example, we consider a tube bounded by a conservative pressure node at the inlet and a dissipative outlet described by the terminal impedance: Zout = γ
ρ f ce
[7.63]
Sf
where γ = 0 stands for a conservative pressure node and γ =1 for an anechoic termination. A pressure source at the inlet is assumed to produce a transient pressure signal at the inlet, which lasts less than the back and forth travelling time Ta = 2 L / ce of the acoustic waves. Solving the problem by using the Laplace transform is straightforward and only the final results need to be presented here. The Laplace transform of the pressure field is found to be: p ( x; s ) = P0 ( s )
e
−
sx ce
⎛ x⎞ − s ⎜ Ta − ⎟ ce ⎠
− Re ⎝ 1 − R e − sTa
[7.64]
where the reflection coefficient is: R=
1− γ 1+ γ
[7.65]
Once more the denominator in [7.64] is expanded as a power series in the exponential delay term, which gives: ⎛ x⎞ ⎛ − sx − s ⎜ Ta − ⎟ ⎞ n c p ( x; s ) = P0 ( s ) ⎜ e ce − Re ⎝ e ⎠ ⎟ 1 + Re − sTa + … + ( Re − sTa ) + … ⎜ ⎟ ⎝ ⎠
(
)
[7.66]
The pressure response in the time domain is therefore composed of two time series of transients homothetic of the source signal. The first series comprises the initial forward travelling wave, which is identical to the source signal, plus the waves reflected at the dissipative outlet. Accordingly, magnitude of the successive waves of the series is decreasing in amplitude as R n (R < 1). The second series comprises all the waves which are reflected at the inlet. As a conservative pressure node is assumed there, the sign of these waves is changed with respect to that of those of the first series. Amplitude is found to decrease as R n +1 . As expected, in the case of an anechoic termination, the response is entirely described by the initial forward wave,
608
Fluid-structure interaction
and, at the opposite, if dissipation is nil, the series of undamped waves last for ever with the same shape as the pressure source signal. It is however necessary to mention that numerical simulation of dissipative terminal impedances gives rise to important difficulties, when performed in the time domain, using either modal or finite element discretization techniques. This is particularly clearly apparent when numerical simulation of an anechoic termination is attempted. The subject is of major interest since in many problems involving wave radiation, the physical domain must be truncated to a manageable finite size to avoid prohibitive computational costs. Thus, artificial boundary conditions at the limits of the computational domain are needed to avoid spurious reflection of the outgoing waves. Various computational techniques have been developed and the subject is still a topical issue. The reader interested into the subject is reported to the specialized literature, the few following references, which are by far not exhaustive, are useful to initiate a bibliographic study [BER 94], [HU 96], [COL 98], [TUR 98], [VAY 02], [GIV 04]. 7.1.3.3 Time domain: dissipative fluid Let us consider for instance the problem of the waves excited by a pressure source P(t) located at the tube inlet, where the terminal impedance is assumed to be a conservative pressure node. Equation [7.22] reads as: ∂2 p ∂p ∂2 p + C + ρ =0 f f ∂x 2 ∂t ∂t 2 p ( 0; t ) = P ( t )
− ρ f ce2
[7.67]
Laplace transforming equation [7.67] gives: d 2 p d 2 p + s C + ρ s p = ⇔ − k 2 ( s ) p = 0 0 ( ) f f dx 2 dx 2 p ( 0; s ) = P ( s )
− ρ f ce2
[7.68]
The Laplace transform of the pressure field is thus of the type: p ( x; s ) = A+ e − kx + A− e + kx
[7.69]
where the wave number k ( s ) is: k (s) =
s (C f + ρ f s )
[7.70]
ρ f ce2
Hence, the damped waves are found to be dispersive. Using the damped momentum equation in [7.21], the Laplace transform of volume velocity is found to be: q ( x; s ) = S f
2 e
ρfc
(
s
(ρ f s + Cf
))
(A e −
+ kx
− A+ e − kx )
[7.71]
Energy dissipation by the fluid
609
Once more, the coefficients A+ and A− are adjusted to the boundary conditions. After a few straightforward manipulations, we arrive at: p ( x; s ) = P ( s )
η ( s ) cosh k ( L − x ) + sinh k ( L − x ) η ( s ) cosh kL + sinh kL
q ( x; s ) = S f P ( s )
s
2 e
ρfc
(( ρ
f
η ( s ) sinh k ( L − x ) + cosh k ( L − x ) η ( s ) cosh kL + sinh kL
))
s + Cf
[7.72]
[7.73]
where η ( s ) is expressed in terms of the outlet impedance Zout ( s ) as: η ( s ) = Zout ( s ) S f
s
2 e
ρfc
(( ρ
f
[7.74]
))
s + Cf
The difficulty now is to revert to the time domain. In general, depending on the outlet impedance, the problem reveals as either very tough or even inextricable. However, with the aim to illustrate the effect of viscous and proportional damping on wave propagation, it is tempting to simplify the problem by considering a tube terminated by an anechoic impedance. The first step is thus to determine the anechoic outlet impedance in the case of damped waves. One expedient way, is to retain the outgoing wave only in the solution [7.69]. It is easy to show that: p + ( x; s ) = A+ e − kx
; q+ ( x; s ) =
(ρ
Sf k f s + Cf
)
A+ e − kx
[7.75]
Hence with the aid of [7.70], the anechoic impedance and η coefficients follow as: Zout =
p + ( L; s ) ce ρ f = q + ( L; s ) Sf
η ( s ) = Zout ( s ) S f
(ρ
f
s +Cf s
s
ρ f ce2
)
(( ρ
f
s +Cf
[7.76]
))
=1
Substituting the particularly simple value η = 1 into [7.72] and [7.73], we arrive at the outgoing wave: −k s x p ( x; s ) = P ( s ) e ( )
q ( x; s ) = S f P ( s )
[7.77] s
2 e
ρfc
(( ρ
f
s + Cf
))
e
−k (s)x
[7.78]
Incidentally, the results [7.77] and [7.78] could have been derived even more directly by adjusting the outgoing wave [7.75] to the boundary condition at the inlet. Nevertheless, the procedure adopted just above presents the interest of emphasizing
610
Fluid-structure interaction
that the anechoic impedance differs depending wether the fluid is itself conservative or dissipative. Notice from [7.74] that if C f vanishes we recover the standard anechoic result Zout = ρ f ce / S f . Returning to the problem of inverting the Laplace transforms [7.77] and/or [7.78], since k(s) is a fairly complicated function of s in comparison with the non dissipative case, inversion is generally untractable analytically, unless suitable forcing functions may be found, which turns out to be the case in particular if a Heaviside step P0 U ( t ) is selected as the time history of the pressure source. The corresponding fields [7.77] and [7.78] can be written as: p ( x; s ) =
x P0 − ce s (C f / ρ f + s ) e s −
x
[7.79]
(
s C f / ρ f +s
)
S P e ce q ( x; s ) = f 0 ρ f ce s ( C / ρ + s ) f f
[7.80]
Analytical inversion of [7.80] is available, though calculation is substantially more difficult to work out than in the problems we presented up to now, because the singularities of the Laplace transforms [7.79] and [7.80] comprise both poles and branching points. The reader interested in the detailed solution can be reported to [STA 70], where the solution is extended to the 2D and 3D cases, and for the French reading persons to [ANG 61] where a particularly enlightening and detailed presentation of the solutions of the so-called equation of telegraphy is presented. The latter deals with the electromagnetic waves which travel along an electrical line including distributed self-induction, capacity and resistance and which is closed by a terminal impedance, matched or not (i.e. in the present context anechoic or not). It includes the damped wave equations [7.22] or [7.67] as a particular case. As expedient shorthand, solution is obtained here by using a table of Laplace transforms (see for instance [ABR 84], [BRO 04] or [ANG 61]) where the following formula is available:
(
⎛ s + β − s s + 2β ) ( ) ⎜ ( TL ⎜ s ( s + 2β ) ⎜ ⎝ −1
⎧ −βt ν ⎪e β ⎨ ⎪ ⎩
ν /2
⎛t −λ ⎞ ⎜ ⎟ ⎝t+λ ⎠
(
2
)
ν
⎞ ⎟ −λ ⎟e ⎟ ⎠
Iν β t − λ
2
)
s( s +2 β )
=
[7.81] if t ≥ λ
0 if t < λ
where Iν stands for the modified Bessel function of the first kind and of order ν. Identifying the coefficients in [7.81] to those of [7.80], the following time history for the volume velocity is obtained:
Energy dissipation by the fluid
q ( x; t ) =
2 ⎛ ⎛ x⎞ ⎞ e− β t I0 ⎜ β t 2 − ⎜ ⎟ ⎟ U ⎜ ρ f ce ⎝ ce ⎠ ⎟⎠ ⎝
S f P0
⎛ x⎞ ⎜t − ⎟ ⎝ ce ⎠
611
[7.82]
where the attenuation or damping coefficient β is defined as: β=
Cf
[7.83]
2ρ f
Using equation [A4-22] in Appendix A4, for large arguments, I 0 (βt) tends asymptotically to the simpler expression: I0 ( β t ) ≅ β t →∞
eβ t 2πβ t
[7.84]
It is also useful recalling that I 0 ( 0 ) = 1 . The pressure field can be related to the volume velocity by using the equation of mass conservation (see [7.21]), which implies: p ( x; t ) = −
ρ f ce2 Sf
t
⌠ ⎮ ⎮ ⎮ ⎮ ⌡0
∂q dθ ∂x
[7.85]
Some care is however needed to perform the derivation of the volume velocity [7.82] with respect to x, because of the finite discontinuity which occurs at the wave front. In fact, the derivative has to be understood in terms of distributions and not in terms of ordinary functions. Therefore, the pressure field is found to be: t
p ( x; t ) = − t
⌠ ⎮ P0 ⎮⎮ e − βθ I 0 ⎮ ⌡0
⌠ ⎮ ⎮ P0 β x ⎮⎮ e − βθ ρ f ce ⎮ ⎮ ⌡0
(β
(
I1 β θ 2 − ( x / ce ) θ 2 − ( x / ce )
2
2
) U ⎛⎜θ − x ⎞⎟ dθ + ⎝
ce ⎠
[7.86]
)
⎛ x⎞ 2 θ 2 − ( x / ce ) δ ⎜ θ − ⎟ dθ ce ⎠ ⎝
And finally as: t ⎧ ⌠ ⎫ 2 ⎮ ⎪ ⎪ I1 β θ 2 − ( x / ce ) ⎮ ⎪P βx ⎪ p ( x; t ) = ⎨ 0 ⎮⎮ e − βθ dθ + P0 e − β x / ce ⎬ U ( t − x / ce ) 2 2 ρ c θ − ( x / ce ) ⎪ f e ⎮ ⎪ ⎮ ⎪⎩ ⎪⎭ ⌡ x / ce
(
)
[7.87]
612
Fluid-structure interaction
Figure 7.18 shows the space profile of the outgoing volume velocity wave for a few values of β, including the conservative case β = 0. It is worth emphasizing that according to [7.82], the front of the wave travels forward at the phase speed ce independently from dissipation.
Figure 7.18. Snapshot of the outgoing volume velocity waves for a few damping values (parameter β) , analytical solution [7.82]
Figure 7.19. Snapshot of the outgoing volume velocity waves for a few damping values (parameter β) , numerical solution by using the modal expansion method
It is of interest to use such an analytical solution as a benchmark case for comparison with numerical methods such as finite elements or modal expansion methods. Selecting here the last approach, it is found convenient to start from the wave equation written in terms of volume velocity for direct comparison with the result [7.82]. It is not difficult to show that the above problem is governed by the forced wave equation: ∂ 2 q 1 ∂ 2 q S f P0 − = δ (t ) ∩ δ ( x ) ∂ x 2 ce2 ∂ t 2 ρ f ce2
[7.88]
Equation [7.88] is then projected onto the modal basis ϕ n( ) = cos ( nπ x / L ) and the q
discretized problem is solved step by step in time by using the Newmark implicit
Energy dissipation by the fluid
613
algorithm corresponding to the mean value of acceleration scheme which is free of numerical damping. Results obtained by retaining the first hundred modes in the modal basis are shown in Figure 7.19, which can be compared to the analytical results of Figure 7.18. Agreement is found to be quite satisfactory, except for the Gibbs oscillations at the wave front of the numerical results which stand as an unavoidable signature of the modal truncation. 7.2. Radiation damping As already illustrated in Chapter 6, when a structure vibrates in contact with another elastic medium, elastic waves are also induced in this second medium. Therefore, if the latter extends to infinity, or is provided with dissipative boundaries, a certain amount of mechanical energy imparted to the structure is radiated away and never returned. As a consequence the structural vibration is damped. Though the physical mechanism responsible for radiation damping is very simple to understand in a qualitative way, the actual computation of the radiated energy is generally by far not a simple task, depending on the geometry and the nature of the waves. In most cases of practical applications, the problem must be solved by using numerical methods, based either on the finite elements or boundary elements methods. 7.2.1
Radiation of acoustic waves
7.2.1.1 Sound intensity and power levels The basic mathematical entities needed to describe the transfer of mechanical energy by wave radiation have been already introduced in Chapter 3, in the context of surface waves. The formulas defining the instantaneous intensity of the wave denoted I ( r ; t ) and the power radiated through a surface (S ) are repeated here for convenience: I ( r ; t ) = p ( r ; t ) X f ( r; t )
[7.89]
p ( r ; t ) is the fluctuating pressure and X f ( r ; t ) is the fluctuating velocity field of
the fluid particles. The energy flux or power radiated through a closed surface (S ) containing the source is defined as: P = ∫∫ I .nd S
[7.90]
(S )
where n is the unit vector normal to
(S )
oriented in the direction of wave
propagation. It is also convenient to relate the pressure field and the normal component of the velocity of the fluid particles through a relation of radiating surface impedance: Z=
p X f .n
[7.91]
614
Fluid-structure interaction
With the aid of [7.89], the instantaneous radiated power is expressed as: PR = ∫∫
(S )
p2 dS Z
The amount of acoustic energy radiated through
[7.92]
(S )
after a time T follows
immediately as: ER = PR T
[7.93]
In the case of periodic, or steady random noise signals, it is useful to define a mean power, averaged on a suitable time T: P R = ∫∫
(S )
p2 dS Z
[7.94]
where p 2 designates the mean square value of the pressure fluctuations, as defined by the averaging integral: p2 =
t +T
1 ⌠⎮ 1 T ⎮⌡t1
p 2 ( t ) dt
[7.95]
Sound extends over a a very large spectral range. In particular, human ear is able to perceive frequencies from about 25 up to 30000 Hz. Sound power and intensity received at a particular place can also vary in very large proportions. The threshold of sound intensity I0 perceptible to human ear is about 10−12 Wm −2 , whereas the threshold for pain and rapid deterioration of the auditory capabilities is of the order of 1 Wm −2 . Therefore logarithmic scales are found to be more appropriate than linear ones to describe such variations. Logarithmic scaling of sound intensity is as follows: ⎛I ⎞ IL = 10log10 ⎜ ⎟ ⎝ I0 ⎠
[7.96]
where I is the sound intensity expressed en Wm −2 , and IL designates the intensity level. IL is expressed in decibels (dB) using as a scaling factor the threshold value I0 = 10−12 Watt/m 2 . Accordingly, the sound intensity corresponding to the pain threshold is 120 dB. Other quantities than sound intensity are of common use in experimental acoustics, as discussed in detail for instance in [BLA 00]. The most commonly measured sound quantity is the fluctuating pressure. Based on its mean square value, the following logarithmic scale is defined: ⎛ p2 ⎞ ⎟ SPL = 10log10 ⎜ ⎜ p2 ⎟ ⎝ 0⎠
[7.97]
Energy dissipation by the fluid
615
where the acronym SPL stands for sound pressure level. The scaling factor p0 of pressure differs according to the nature of fluid; in a gas p0 = 20μ Pa whereas in a liquid p0 = 1μ Pa . The value adopted as a reference for gases corresponds approximately to the audibility threshold of a young person in the frequency range 1-4 kHz, which in turns corresponds to the maximum of auditory sensivity of human ear. As typical orders of magnitude, the SPL of normal conversation is 74 dB and that of a jet taking off is about 120 dB. On the other hand, it is useful recalling that if r.m.s pressure is changed by a factor two, the corresponding SPL is changed by about 6 dB ( log10 4 6.021 ). Sound transmission is generally specified in terms of sound power. The level of sound power (PWL) is defined as: ⎛P ⎞ PWL = 10log10 ⎜ ⎟ ⎝ P0 ⎠
[7.98]
P is the sound power transmitted through a given surface and P0 is the scaling
factor set conventionally to P0 = 10−12 W . As typical orders of magnitude, a very soft whisper may be as low as 30 dB ( 10−9 W ), while for a space launcher it can be as large as 195 dB (30 MW). The quantities defined just above to describe sound radiation can also be particularized to harmonic waves, either of the travelling or the standing type. In a harmonic wave, pressure and velocity of the particles are described by the complex amplitudes, written here as: p = Peiωt
;
u = Ue (
i ω t +ψ )
[7.99]
where P designates the absolute value of pressure and U that of the velocity of the particles. The phase angle of velocity with respect to pressure is denoted ψ. Once more, starting from the complex amplitudes [7.99], the quantities related to the real motion of the fluid are recovered by considering either the real or the imaginary part of the complex quantities. The important point here is that the impedance, as defined in the spectral domain, is generally complex: Z=
p = ZR + iZI = Z e − iψ u
[7.100]
As a consequence, the mean value of sound intensity as averaged on a period, is written as: ω I (r ) = n 2π
Whence:
2π
⌠ω ⎮ ⎮ ⌡0
Re (Z u ) Re ( u ) dt
[7.101]
616
Fluid-structure interaction
ω I (r ) = I (r ) = 2π
2π
⌠ ω ⎮ ⎮ ⌡0
Z U 2 cos(ω t −ψ ) cos ωt dt
[7.102]
and finally: Z U 2 cosψ p2 cosψ = Re (Z ) u 2 = I (r ) = 2 Z
[7.103]
As a particular case, if the impedance is purely imaginary ψ = π / 2 hence no energy is radiated; which is a natural result as it corresponds to the conservative case of standing waves. On the other hand, it is noted that such a result may be considered as the mechanical analogue of that which holds in electromagnetism, according to which the amount of mean electrical power which is dissipated is equal to the electrical resistance times the mean square value of the electrical intensity or the mean square value of the electrical voltage devided by the resistance. Electrical resistance stands precisely for the real part of impedance, which characterizes dissipation. 7.2.1.2 Piston-fluid column system: motion of the piston
Figure 7.20. Rigid tube containing a fluid column and coupled to a mass-spring system
Let us consider the simplest possible problem to deal with, that is once more, the piston-fluid column system shown in Figure 7.20. It was already analysed in Chapter 6 subsections 6.2 and 6.5, where fluid compressibility was accounted for but no dissipation was assumed to occur in the fluid. Here, interest is more particularly concentrated on the motion of the piston and next subsection will concentrate on the fluid oscillations. If the piston is driven by a harmonic force, harmonic sound waves of the same frequency are excited in the fluid column. Furthermore at the open outlet, pressure is not exactly zero in such a way that some amount of the internal sound energy is radiated away. The energy such extracted from the vibroacoustic system can be accounted for by a dissipative terminal impedance. Analytical formulation of the real and imaginary parts of this kind of impedance is postponed to subsection 7.2.1.4. Here, an impedance of the type [7.63]
Energy dissipation by the fluid
617
is assumed for sake of simplicity in introducing the subject. Therefore, the dimensionless outlet impedance verifies the relation: tan αout = −iγ
[7.104]
Substituting [7.104] into equation [6.79], where structural damping is set to zero, we obtain the dimensionless equation: ⎛ 2 2 ⎜ κ 0 − κ − κμ f ⎝
⎛ tan κ − iγ ⎜ ⎝ 1 + iγ tan κ
⎞⎞ Xs =1 ⎟⎟ ⎠⎠ X0
[7.105]
where the following scaling factors and reduced quantities are defined: ω0 =
Ks Ms
;
X0 =
F0 Ks
; κ=
ωL ce
; κ0 =
ω0 L ce
; μf =
Mf Ms
[7.106]
It may be noted that the dissipative term is rather involved and cannot be reduced to viscous damping except in the particular case of anechoic impedance, in which case equation [7.105] simplifies into:
(κ
2 0
− κ 2 + iκμ f
) XX
s
=1
[7.107]
0
Figure 7.21. Vibroacoustic response spectrum of the piston including radiation damping
As illustrated in Figure 7.21, the response spectrum of the piston displacement is marked by a series of resonant peaks. In Chapter 6 it was shown that the first resonance corresponds to a strongly coupled vibroacoustic mode, whereas the higher resonances are dominated by the acoustical part of the system. Therefore, the
618
Fluid-structure interaction
equivalent damping ratio, characterized by the relative width of the resonance peak, is much less for the first mode than for the others.
Figure 7.22. Equivalent viscous damping ratios of the two first vibroacoustic resonances
7.2.1.3 Piston-fluid column system: acoustic waves With the aid of the results [6.11] which specifies the pressure field related to the displacement X s of the piston and [7.105] which relates X s to the harmonic exciting force exerted on the piston, the acoustic pressure field forced in the fluid column can be written as: ⎛ μ κ F ⎞⎛ ⎞ sin κξ − tan (κ + αout ) cos κξ P (ξ ; κ ) = ⎜ f 0 ⎟ ⎜ 2 ⎜ S ⎟ ⎜ κ − κ 2 + 2iκ κς − κμ tan (κ + α ) ⎟⎟ f 0 0 f out ⎠ ⎝ ⎠⎝ 0
[7.108]
where the viscous damping ratio ς 0 given by [6.80] is included into the model to account for structural dissipation. Considering first the case of conservative terminal impedance, a pressure node for instance, the complex pressure field can be expressed as: ⎞ ⎛ μ κ F ⎞⎛ sin κ (1 − ξ ) eiκθ ⎟ p (ξ ; κ , θ ) = ⎜ f 0 ⎟ ⎜ 2 2 ⎜ S ⎟ ⎜ cos κ (κ − κ + 2iκ κς ) − κμ sin κ ⎟ f f 0 0 0 ⎝ ⎠⎝ ⎠
[7.109]
where the dimensionless time θ is defined as θ = tce / L . The particle fluid velocity is in phase quadrature with the pressure, therefore ψ = π / 2 and the sound intensity [7.103] is zero as should be, since standing waves radiate no energy. This is not to say that acoustical energy in the fluid column is zero. Instantaneous mechanical energy is obtained by adding the potential and kinetic energies: L
E (θ )m =
Sf 2
⌠ ⎮ ⎮ ⎮ ⎮ ⎮ ⌡0
( Real ( p ) ) dx + S 2
2 e
ρfc
⌠
f ⎮
2
L
⎮ ⎮ ⌡0
ρ f ( Real ( u ) ) dx 2
[7.110]
Energy dissipation by the fluid
619
Calculation is straightforward, however rather tedious, though the final result is simple. To alleviate notation, formula [7.109] is written in the condensed form: ⎛ μ κF ⎞ p (ξ ; κ , θ ) = ⎜ f 0 ⎟ ( a (κ ) + ib (κ ) ) sin κ (1 − ξ ) eiκθ ⎜ S ⎟ f ⎝ ⎠
[7.111]
where a and b are: a (κ ) = b (κ ) =
( cos κ (κ ( cos κ (κ
cos κ (κ 02 − κ 2 ) − κμ f sin κ 2 0
− κ 2 ) − κμ f sin κ
)
2
+ ( 2κ 0κς 0 cos κ )
2
[7.112]
−2κ 0ς 0κ cos κ
2 0
− κ 2 ) − κμ f sin κ
)
2
+ ( 2κ 0κς 0 cos κ )
2
The particle velocity follows as: ⎛ μ κ ⎞⎛ F ⎞ u (ξ ; κ , θ ) = ⎜ f ⎟ ⎜ 0 ⎟ ( b (κ ) − ia (κ ) ) cos κ (1 − ξ ) eiκθ ⎜ ρ c ⎟⎜ S ⎟ ⎝ f e ⎠⎝ f ⎠
[7.113]
Selecting the real parts of [7.111] and [7.113] to describe the real vibration and substituting the results into the energy equation [7.110], the following expressions of elastic and kinetic energy are obtained after a few simple manipulations: Ee (θ ) = Eκ (θ ) =
μ 2f κ 2 F02 L 2 ρ f ce2S f 2 f
2
2 0
μ κ F L 2 ρ f ce2S f
(a
2
⌠
1
cos2 κθ + b2 sin 2 κθ − ab sin ( 2κθ ) ) ⎮⎮ sin 2 (κ (1 − ξ ) )d ξ ⌡0 ⌠
1
( b2 cos2 κθ + a 2 sin2 κθ + ab sin ( 2κθ ) ) ⎮⎮ cos2 (κ (1 − ξ ) )dξ ⌡0
[7.114] Hence, as could have been anticipated, mechanical energy is independent of time. It is written as: Em =
μ 2f κ 2 F02 L ( a 2 + b2 ) 2 ρ f ce2S f
(
=
μ 2f κ 2 F02 L
2 ρ f ce2S f cos κ (κ 02 − κ 2 ) − κμ f sin κ
[7.115]
)
2
+ ( 2κ 0κς 0 cos κ )
2
620
Fluid-structure interaction
Figure 7.23. Power spectrum density of sound pressure at tube inlet, ς 0 = 1% and lossless fluid and outlet ( M s = 10−3 kg , M f / M s = 0.008 , f 0 = 250 Hz , F0 = 1 N )
As an application, we consider a circular cylindrical tube, length L = 34 cm, radius R = 2.5 mm, filled with air at STP ce = 340 m/s and ρ f = 1.2 kg / m 3 . In vacuum, the natural frequency of the mass-spring system is set to 250 Hz, which corresponds to the quarter wavelength acoustical resonance. The damping ratio is set to ς 0 = 0.01 . The mass of the piston is M s = 10−3 kg and the fluid added mass ratio is M f / M s = μ f 0.008 . Magnitude of the harmonic driving force is F0 = 1N . Figure 7.23 shows the spectral density of pressure at the tube inlet where the fluidstructure interface is located. The plot at the left is covering a frequency range which is sufficiently broad to put in evidence the nature of the successive peaks of response. The stars indicate the frequencies of the acoustic modes for a tube closed at one end and open at the other. The response peaks are precisely centered at such frequencies, except the first two peaks lying in the close vicinity of the first acoustical mode, which means that even if μ f is as low as 0.008, vibroacoustic coupling is still clearly discernable, as better seen in the right-hand plot. That such peaks correspond to vibroacoustic modes with significant displacement of the piston is also corroborated by their relatively large width in comparison with the higher resonances. Actually, as in the present model both fluid and terminal impedances are supposed lossless, the acoustical resonances with negligible motion of the piston are practically undamped, while the vibroacoustic resonances are damped due to dissipation related to the motion of the piston. Incidentally, detailed analysis shows that damping ratio of the two vibroacoustic peaks is precisely ς 0 / 2 = 0.5% . Finally, it may be worth mentioning that if frequency f 0 of the excitation force is distinct from any acoustical resonance, the response spectrum consists of a series of very narrow lines at the acoustical frequencies plus a relatively broad peak centred at f 0 . If the tube outlet is terminated by the dissipative impedance [7.104], the velocity of the fluid particles is related there by the relation:
Energy dissipation by the fluid
p = i ρ f ce tan α out u = ρ f ceγ u
621
[7.116]
Therefore p and u are in phase which means that some amount of the acoustical energy contained within the tube is radiated away through the outlet surface. The corresponding sound power is determined by using equation [7.104]: P =
S f p 2 (ξ = 1)
[7.117]
ρ f ceγ
The pressure field [7.108] becomes: ⎞ ⎛ μ κ F ⎞⎛ (1 + iγ tan κ ) sin κξ − ( tan κ − iγ ) cos κξ ⎟ P (ξ ; κ ) = ⎜ f 0 ⎟ ⎜ ⎜ S ⎟ ⎜ (1 + iγ tan κ ) (κ 2 − κ 2 + 2iκ κς ) − κμ ( tan κ − iγ ) ⎟ f 0 0 0 f ⎝ ⎠⎝ ⎠
[7.118]
With the aid of a few trigonometric manipulations the complex field [7.118] can be written as: ⎛ μ κ F ⎞ ⎛ ( sin κ (ξ − 1) + iγ cos κ (ξ − 1) ) (η1 − iγη2 ) ⎞ P (ξ ; κ ) = ⎜ f 0 ⎟ ⎜ ⎟ ⎜ S ⎟⎜ ⎟ η12 + γ 2η22 f ⎝ ⎠⎝ ⎠
[7.119]
where the coefficients η1 and η2 are calculated by neglecting the structural damping to alleviate algebra. They are thus defined as: η1 = cos κ (κ 02 − κ 2 ) − κμ f sin κ
; η2 = sin κ (κ 02 − κ 2 ) + κμ f cos κ
[7.120]
The complex amplitude of the sound pressure at the tube outlet is: ⎛ μ κ F ⎞ ⎛ γ (γη + iη ) ⎞ P (ξ ; κ ) = ⎜ f 0 ⎟ ⎜ 2 2 2 21 ⎟ eiθ ⎜ S ⎟ η +γ η f 2 ⎠ ⎝ ⎠⎝ 1
[7.121]
Taking the real part of [7.121] we obtained the real pressure field: p (1; κ ;θ ) =
γμ f κ F0 (γη2 cos θ − iη1 sin θ ) S f (η12 + γ 2η22 )
[7.122]
Time averaging the real pressure squared is immediate and finally, the sound power radiated at the tube outlet is found to be given by the remarkably simple formula:
(μ κ F ) P = f
2
0
2S f ρ f ce
γ
[7.123]
622
Fluid-structure interaction
Figure 7.24. Power spectrum density of sound pressure at tube inlet,lossless fluid and dissipative outlet γ = 0.1 , M f / M s = 0.008 , f 0 = 250 Hz , F0 = 1 N
As an application, we consider the same circular cylindrical tube as above. Here, it is provided with a dissipative node γ = 0.1 at the outlet. Figure 7.24 displays the spectral density of the pressure at the tube inlet for comparison with Figure 7.23. The major difference between them lies in the source of dissipation. Contrasting with the spectra of Figure 7.23, those of Figure 7.24 which correspond to a significant motion of fluid are also marked by relatively broad peaks. As illustrated in the lower plots, if f 0 is distant from any acoustical resonance, the corresponding peak is practically undamped, hence shaped as a very narrow line. Figure 7.25 shows the acoustical power radiated outside versus frequency. It is of interest to note that the device acts as a powerful source of sound, whose efficiency increases as the square of frequency and in proportion with γ.
Energy dissipation by the fluid
623
Figure 7.25. Sound power radiated outside γ = 0.1 , M f / M s = 0.008 , f 0 = 250 Hz ,
F0 = 1 N
Figure 7.26. Power spectrum density of sound pressure at tube inlet, ς 0 = 0 , lossless fluid and anechoic outlet γ = 1 , M f / M s = 0.008 , f 0 = 250 Hz , F0 = 1 N
Finally, Figure 7.26 refers to the extreme case of an anechoic termination. As already shown in subsection 6.1 the system behaves as a single DOF system, since no acoustic resonance can occur in the absence of back reflected waves. On the other hand, though all the acoustical energy is dissipated outside, the damping ratio, based on the half power widh of the peak of the vibroacoustic mode remains fairly small ς 1 0.003 , as indicated in the right-hand side plot. This is clearly due to the very small value of the mass ratio μ f in air. Indeed as μ f increases, so does coupling; as a consequence the frequency splitting of the two vibroacoustic modes is enlarged and so is the loss factor related to sound radiation. However, the terminal impedance of the open end of a tube differs, in reality, from the simplistic model described here. This important point is the object of the next subsection.
624
Fluid-structure interaction
7.2.1.4 Piston-fluid system: terminal impedance for an open tube As already described in Chapter 2 subsection 2.2.2.4, in the context of modelling fluid inertia through a circular hole, an open tube termination can not correspond to a pressure node since some amount of fluid oscillates through the hole. The problem of determining the terminal impedance which holds for the internal plane waves can be tackled by modelling the fluid through the orifice as an equivalent oscillating plug of fluid, or piston, which radiates sound outside the tube. Sound radiation by an oscillating circular piston was already described in Chapter 5 subsections 5.3.4.7 to 5.3.4.9, where it was shown in particular that two basic configurations have to be distinguished, namely the baffled one and the unbaffled one. Such a distinction holds also in the case of the open tube end, which can be baffled or not, as sketched in Figure 7.27 for a circular cylindrical tube. To characterize the orifice as an impedance seen by the harmonic plane waves inside the tube, it is appropriate to deal with the mean sound pressure denoted p and particle velocity denoted u , as averaged over the surface of the orifice. Thus the impedance is defined as: Z=
p F = u π R02U
[7.124]
where U designates the velocity of the equivalent piston and F the resulting sound pressure force exerted on it. To determine F, it is needed to integrate the sound pressure at each point on the disk over the whole area of the disk. On the other hand, considering the baffled configuration, the pressure field at a point of the orifice can be calculated by using the Kirchhoff-Helmholtz integral [5.302]. Noting that in the present application r lies in the plane of the tube outlet, grad r −1 .n is necessarily zero and formula [5.302] simplifies conveniently into: −iωρ f U p( r ; t ) = 2π
⌠ ⎮ ⎮ ⎮ ⎮ ⌡(S f
)
−ikP0 e − ikr dS = r 2π
⌠ ⎮ ⎮ ⎮ ⎮ ⌡(S f
)
e − ikr dS r
[7.125]
Now, to calculate the sound pressure force, one has to solve a quadruple integral. For that purpose, it seems advisable, a priori at least, to adopt polar coordinates with the pole at the centre O of the orifice. As expressed in terms of the polar coordinates ( r,θ ) of an emiting point and those of a receiving point ( r ′,θ ′) , the integral [7.125] is first written as:
F=
⌠ ⎮ ⌠ ⎮ −ikP0 ⎮⎮ d S ′⎮ ⎮ ⎮ 2π ⎮ ⎮ ⌡(S f ⎮ ⌡(S f′ )
)
−ikP0 e − ikr dS = r 2π
2π
⌠ ⎮ ⎮ ⎮ ⌡0
R
⌠ 0 dθ ′⎮⎮ ⎮ ⌡0
⌠
θ′
⌠
r ' dr ' ⎮⎮ dθ ⎮⎮ ⌡0
r′
⌡0
e − ikr rdr r
[7.126]
Energy dissipation by the fluid
625
Figure 7.27. Sound radiation through the open end of a tube
Analytical integration of [7.126] is far from easy. The problem was solved by Lord Rayleigh. The method, described in particular in [RAY 98], [ROC 71], [TEM 01], [BLA 00], is so ingeneous that it deserves to be repeated here. The trick is to restate the problem by considering a receiving point P at the distance R from O lying within the range 0 ≤ R ≤ R0 . In a first step, partial integration is carried out to produce the pressure p ( R ) at P which results from the contribution of all the points
lying in the circular disk of centre O and radius R, see Figure 7.28. The distance h between P and the current emissive point P′ is first varied between 0 and H = 2 R cos φ , where φ designates the angle beween the diameter passing through P and the straight line PP′ . Sweeping of the whole circular disk is then achieved by letting h to vary from zero to 2 R cos φ and φ from −π / 2 to +π / 2 . Accordingly, integration is performed by using the polar coordinates h and φ , centred at the receiving point P. Hence, in agreement with the impedance relation [7.124]: p ( R; t ) =
−ikP0 2π
+π / 2
⌠ ⎮ ⎮ ⎮ ⌡− π / 2
⌠
dφ ⎮⎮
2 R cos φ
⌡0
+π / 2 ⌠ ⎞ e -ikh P ⎛ hdhdφ = 0 ⎜ π − 2⎮⎮ e-ik 2 R cosφ dφ ⎟ 2π ⎝ h ⌡0 ⎠
[7.127]
626
Fluid-structure interaction
Figure 7.28. Variables and integration path to compute the pressure force
With the aid of formula [5.248], p( R; t ) is expressed analytically as: p ( R; t ) =
P0 (1 − J 0 ( 2kR ) + iK0 ( 2kR ) ) 2
[7.128]
where: +π / 2
J 0 ( 2kR ) =
2 ⌠⎮ π ⎮⌡0
cos ( 2kR cos φ ) d φ
[7.129]
+π / 2
2⌠ K 0 ( 2kR ) = ⎮⎮ π ⌡0
sin ( 2kR cos φ ) d φ
Let us mention for reference that K 0 is named a Struve function of zero order.
Finally, to complete the solution of the quadruple integral [7.126], p ( R; t ) is integrated over the whole surface of the orifice, giving thus the resulting force: ⌠
2π
⌠
R0
⌠
F = ⎮ dθ ′ ⎮⎮ p ( kr' ) r'dr' = π P0 ⎮⎮ ⌡0
⌡0
R0
⌡0
(1 − J ( 2kr ′) + iK ( 2kr ′) ) r ' dr ' 0
0
[7.130]
The remaining integration can be solved analytically by using the recurrence relations [A4.8] of Appendix A4 which hold for the Struve as well as for the Bessel functions. One finds in [BLA 01]: ⎛ 2 J ( 2kR0 ) 2 K1 ( 2kR0 ) ⎞ F = π P0 R02 ⎜ 1 − 1 +i ⎟ 2kR0 2kR0 ⎝ ⎠
[7.131]
In [MOR 86] and [ROC 71], the imaginary part of [7.131] is expressed as: K1 ( 2kR0 ) 4 ⌠ +π / 2 sin ( 2kR cos φ ) sin 2 φ d φ = ⎮⎮ 2kR0 π ⌡0
[7.132]
Energy dissipation by the fluid
627
Finally, the impedance of the open end for a circular cylindical tube is found to be: Z=
K ( 2kR0 ) ⎞ ⎛ J ( 2kR0 ) F = ρ f ce ⎜ 1 − 1 +i 1 ⎟ 2 π R0 U kR0 kR0 ⎝ ⎠
[7.133]
As could have been anticipated, the impedance is complex. The real part accounts for sound radiation outside the tube and the imaginary part accounts for the conservative aspects of the fluid oscillation in the vicinity of the orifice, which reduce to an added fluid inertia in the low frequency range.
Figure 7.29. Real and imaginary parts of the acoustic impedance of the tube open end, referred to the specific impedance of the fluid
Figure 7.29 is a plot of the real and imaginary parts of the terminal impedance normalized to the specific impedance of the fluid inside the tube. Once more, the nondimensional number κ = kR0 can be rightly interpreted as the dimension of the emissive object normalized to the sound wavelength. It can be verified that if κ is sufficiently large, the impedance is essentially real and equal to the specific impedance; which means that the tube end is practically anechoic (γ = 1) if its size largely exceeds the sound wavelength. Said differently, sound propagates essentially as in a free medium. However such an asymptotic behaviour is by far not valid for the plane wave approximation inside the tube which corresponds precisely to the range κ < kR0 . In the large wavelength range, the imaginary part of the terminal impedance largely prevails on the real part, which means that sound radiation through the open end is poor. In fact, if κ << 1 fluid motion in the vicinity of the orifice is practically incompressible. Consequently, fluid oscillation induces an
628
Fluid-structure interaction
inertia effect, as already seen in Chapter 2 and a very little amount, if any, of mechanical energy delivered to the fluid column is spent in acoustic radiation. Referring now to the case of the axial pressure field induced by a vibrating piston, it may be said that in the long wavelength range the pressure field induced at the open end of the tube is essentially of the near field and incompressible type, while it becomes essentially of the far field and acoustical type in the range of short wavelengths, which extend beyond the plane wave approximation in ducts. Such a separation into a near and a far field component also holds in other geometries as further illustrated in the case of spherical and cylindrical waves which make the object of next two subsections. Returning to the nearly incompressible case, the impedance can be simplified by expanding in Taylor series to the first order in kR0 either the Bessel and Struve functions in [7.133] or the kernel of the integral [7.125]. Adopting the second possibility, we obtain: −ikP0 p( r1 ; k ) = 2π
⌠ ⎮ ⎮ ⎮ ⎮ ⌡(S f )
dS k 2 P0 − 2π r
⌠ ⎮ dS ⎮ ⎮ ⌡(S f )
[7.134]
The real part of [7.134] gives the radiation part of the terminal impedance: Re (Z ) =
− ρ f ce ( kR0 )
2
[7.135]
2
The imaginary part gives the inertial part of it: Im (Z ) =
−k ρ f ce 2π
⌠ ⎮ ⎮ ⎮ ⎮ ⌡(S f )
dS r
[7.136]
The remaining integral can be solved by using the same Rayleigh’s trick as for the integral [7.126]. The result is: Im (Z ) = − ρ f ce
8kR0 3π
[7.137]
It is suitable to collect the results [7.135] and [7.137] to express the boundary condition to be fullfiled by the plane waves at the open end of a tube as the terminal complex impedance: 2 2 ⎛ 8kR ( kR0 ) ⎞ 8kR0 ⎞ +i ρ f ce Z ρ f ce ⎛ ( kR0 ) 0 tan α out ⇒ tan α out = ⎜ = +i +i ⎜− ⎟= ⎟ ⎜ 3π 2 3π ⎟⎠ 2 ⎟⎠ Sf S f ⎜⎝ Sf ⎝ [7.138]
Furthermore, the coefficients are sufficiently small to assimilate the tangent to its argument, whence the dimensionless impedance:
Energy dissipation by the fluid
αout =
( kR0 ) 8kR0 +i 3π 2
629
2
[7.139]
According to the impedance [7.139], the natural frequencies are slightly lower than the values predicted by neglecting the real part of [7.139]. The correction can be accounted for by increasing the actual length of the tube by the small amount: ΔL =
8R0 0.85R0 3π
[7.140]
Incidentally if the tube is not flanged, the corrective length is smaller (see for instance BLA [00], [AND 70] and [LEV 48]): Δ L 0.6133R0
[7.141]
Radiation damping is: ςn
nπ 2
⎛ R0 ⎞ ⎜ ⎟ ⎝ L⎠
2
[7.142]
7.2.1.5 Spherical shell pulsating in an infinite medium Let a spherical shell of radius R0 be immersed in an infinite extent of fluid. As sketched in Figure 7.30, the shell is assumed to vibrate according to the breathing mode with the amplitude U 0 . In vacuum the corresponding modal oscillator is governed by the equation of harmonic and undamped oscillator: ⎡⎣ K 0 − ω02 M 0 ⎤⎦U 0 = 0
[7.143]
K 0 and M 0 are the generalized stiffness and mass coefficients associated to the breathing mode. The pressure field induced in the fluid by such a vibration is governed by the equations: ⎧ d 2 p 2 dp ⎛ ω ⎞ 2 ⎪ 2 + +⎜ ⎟ p =0 r dr ⎜⎝ c f ⎟⎠ ⎪ dr ⎨ dp ⎪ = ω 2 ρ f U0 ⎪ dr Ro ⎩
[7.144]
The general solution of the differential equation [7.144] is: ⎛ ⎛ r ⎞ r ⎞ ⎡ iω ⎜ t + ⎟ ⎤ ⎜ cf ⎟ 1 ⎢ iω ⎜⎜⎝ t − c f ⎟⎟⎠ ⎝ ⎠⎥ + A− e ∀ r ≥ R0 p ( r; t ) = A+ e ⎥ r⎢ ⎣ ⎦
[7.145]
630
Fluid-structure interaction
Figure 7.30. Sphere vibrating according to the breathing mode in an infinite extent of fluid
The result [7.145] can be presented as a particular case of the spherical waves described in Chapter 5 subsection 5.1.2.3 in the context of standing waves, or more directly in subsection 5.3.4.1, in the context of the Green function in a 3D unbounded medium, which is closely related to the present problem. It is recalled that the wave A+ is the diverging, or outgoing wave travelling from the surface of the sphere to infinity, and the wave A− is the converging, or incoming wave which travels from infinity toward the sphere surface. In the same manner as in subsection 5.3.4.1, the anechoic condition at infinity implies: A− = 0
[7.146]
The amplitude of the outgoing wave is set to the appropriate value by using the condition at the fluid-structure interface, which gives: ⎛ ⎞ ⎛ r−R ⎞ ⎜ −ω 2 ρ R 2U ⎟ 1 iω ⎜⎜⎜ t − c 0 ⎟⎟⎟ f ⎠ f 0 0 ⎟ e ⎝ p ( r; t ) = ⎜ i R ω ⎜ 1+ ⎟r 0 ⎜ ⎟ c f ⎝ ⎠
∀r ≥ R0
[7.147]
A result which is rewritten in the equivalent form: p ( r; θ ) =
−ω 2 ρ f R02U 0 1 + ( kR0 )
2
(1 − ikR0 )
1 iωθ e U (θ ) r
∀θ ≥ 0
[7.148]
where k = ω / c f is the wave number and θ the retarded time due to propagation of the wave:
Energy dissipation by the fluid
θ =t−
r − R0 cf
631
[7.149]
It is of interest to bring out the similarity between the waves [7.148] and those produced by a monopole source of unit magnitude, concentrated at the centre of the sphere. As seen earlier in subsection 5.3.4.1, the latter stands for the Green function in a 3D unbounded and homogeneous medium, whose Fourier transform can be inferred from equation [5.201] as: ρ G ± ( r; ω ) = f e − iωθ 4π r
[7.150]
Comparison of the waves [7.148] and [7.150] indicates that in the low frequency or small wave number range, kR0 1 the pulsating sphere is practically equivalent to a concentrated volume velocity or monopole source of magnitude: Q = iω 4π R02U 0 = iω S0U 0
[7.151]
where S0 = 4πR02 is the sphere surface area. Such a result is not so surprising, as it merely states that the physical size of the emissive sphere is of no practical importance, except of course as the magnitude of the source is concerned, provided the emitted wavelengths are much larger than R0 . Turning now our attention onto the generalized force exerted by the fluid on the spherical shell, it is immediate to show that the related functional is: F0 ,U 0 = − p ( R0 ; ω ) S0U 0 = ω 2 4πρ f R03
1 − ik R0 1 + ( k R0 )
2
U 02
[7.152]
The vibroacoustic equation of the shell projected on the breathing mode is thus: ⎡ ⎛ 1 − ik R0 ⎞ ⎤ ⎢ K 0 − ω 2 ⎜ M 0 + 4πρ f R03 ⎟⎥U0 = 0 2 ⎜ 1 + ( k R0 ) ⎠⎟ ⎥⎦ ⎢⎣ ⎝
[7.153]
In the range kR0 << 1, the vibroacoustic force can be simplified by expanding [7.152] to the first order in kR0 . The result is the equation of a damped harmonic oscillator: ⎡⎣ K 0 + iωCR − ω 2 ( M 0 + M a ) ⎤⎦ U 0 = 0
[7.154]
where the added mass coefficient is that already calculated in Chapter 2: M a = 4π R03 ρ f
[7.155]
The damping coefficient due to sound radiation is: CR = ω M a kR0
[7.156]
632
Fluid-structure interaction
Since CR is found to depend on frequency, it becomes obvious that energy dissipation through sound radiation by a sphere does not correspond physically to a viscous damping mechanism. However, as long as rate of energy dissipation is small, one can safely define an equivalent viscous damping coefficient by setting ω to the resonant value: ⎛ω R ⎞ CR = ω1 M a ⎜ 1 0 ⎟ ⎝ c0 ⎠
[7.157]
where ω1 = K 0 / ( M 0 + M a ) is the natural frequency of the breathing mode in fluid. The equivalent modal damping ratio follows as: ζ1 =
CR M a k1 R0 = 2 ( M 0 + M a ) ω1 2 ( M 0 + M a )
[7.158]
which is found to be proportional to the modal compressibility factor, or dimensionless wave number k1 R0 , where the pertinent length scale is found to be the radius of the sphere. To investigate the whole range of possible k1 R0 values, it is useful to proceed as in the case of the circular piston or open tube end, by defining the acoustical impedance presented by the fluid to the solid. With the aid of relations [7.148] and [7.151], we obtain: ρ c ( kR0 ) + ikR0 p ( R0 ) = f f 2 2 iω 4π R0 U 0 S0 1 + ( kR0 ) 2
Z=
[7.159]
The analytical form of the impedance [7.159] is significantly simpler than the form [7.133] which holds for a baffled piston. Qualitatively, the behaviour is however the same, as illustrated in Figure 7.31 which superposes the real and imaginary parts of Z for a baffled circular and rigid piston (dashed lines), and for a sphere vibrating according to the breathing mode (full lines). It can be noted that both sets of curves present strong similarities. The imaginary part of the impedance is related to the inertia effect which is largely predominant in the long wavelength range, while the real part of it is related to radiation damping which prevails in the range of short wavelengths. As an asymptotic result, if kR0 tends to infinity, the impedance tends to the anechoic value, which means that the vibroacoustic force is purely dissipative, or, in other terms, that all the mechanical energy delivered to the vibrating solid is transmitted to the fluid as outgoing sound waves. As a consequence of important interest in practice, efficiency of a vibrating solid to radiate sound at a given wavelength can be significantly enhanced by increasing its size, in the range kR0 less than a few units at least. By increasing further R0 the efficiency remains almost the same but emitted sound power is still increased proportionally to the emissive area S0 .
Energy dissipation by the fluid
633
Figure 7.31. Real and imaginary parts of the acoustic impedance of the sphere in breathing mode as referred to the specific impedance of the fluid (full lines: sphere, dashed lines: circular baffled piston)
Substituting the generalized force [7.152] into the equation of the modal oscillator [7.153], the following modal equation of the vibroacoustic system is obtained: ⎛ ⎛ ω R ⎞2 ⎞ ⎡ ⎛ ω R ⎞2 ⎤ ⎛ ω R0 ⎞ 2 2 0 0 K0 ⎜1 + ⎜ ⎟⎟ ⎟ − ω ⎢1 + ⎜⎜ ⎟⎟ ⎥ M 0 − M aω ⎜⎜ 1 − i ⎟=0 ⎜ c c c f ⎟⎠ ⎜ ⎟ ⎢ ⎥ ⎝ ⎝ ⎝ f ⎠ ⎠ ⎣ ⎝ f ⎠ ⎦
[7.160]
which is written in a dimensionless form as: ⎛ 1 − iκ ⎞ ⎟=0 κ 02 − κ 2 ⎜ 1 + μ f 2 ⎜ ⎟ 1 κ + ( ) ⎝ ⎠
[7.161]
where the following dimensionless quantities are defined: κ=
ω R0 cf
; κ0 =
ω0 R0 cf
; ω0 =
K0 M0
; μf =
Ma M0
[7.162]
Such a coupled system is still described by a single degree of freedom, even if the structural breathing mode of the shell is coupled to a compressible fluid. The reason is simple. Since no acoustic modes can exist in a fluid which extends to infinity, there is no mode coupling between the structure and the fluid. Consequently, the number of degrees of freedom required to describe the motion of the solid remains the same as in vacuum, in agreement with the piston-fluid system described in subsection 7.2.1.2, which was found to present infinitely many resonances if some amount of wave reflection occurs at the tube ends and a single resonance only if the
634
Fluid-structure interaction
tube is terminated by an anechoic impedance. Moreover, as in the case of the pistonfluid system, the oscillator equation [7.160] differs from the canonical form which holds for a damped harmonic oscillator. As a consequence, the peak of resonance of the spherical shell vibrating according to the breathing mode in a dense fluid like water, can differ significantly from that predicted by the classical viscous model.
Figure 7.32a. Resonance peak of the spherical shell pulsating according to the breathing mode in water M 0 = 10 kg , R0 = 0.5 m ρ f = 1000 kg c f = 1000 m/s . Full line: radiation damping, circles: equivalent viscous damping. Damping ratios are evaluated by using the half-power bandwitdh method
Discrepancy depends of course of the reduced wave number kR0 and mass ratio μ f , as illustrated in Figures 7.32a and 7.32b. In particular, in air even if kR0 is fairly large, damping remains relatively small due to the small value of μ f . However, the equivalent viscous model is still largely deficient concerning both the resonance frequency and damping. However, since damping is small, the resonance peaks can still be fitted to a viscous model by selecting the ad hoc parameters. Such a data reduction is often used in experimental modal testing using sophisticated data fitting procedures; see for instance [EWI 00].
Figure 7.32b. Resonance peak of the spherical shell pulsating according to the breathing mode in air M 0 = 10 kg , R0 = 0.5 m ρ f = 1.2 kg c f = 340 m/s . Damping ratios are evaluated by using the half-power bandwitdh method
Energy dissipation by the fluid
635
The physical reason underlying such behaviour of radiation damping is again that the force exerted by a compressible fluid on a structure cannot be separated into a combination of inertial, elastic and viscous damping components, except in the asymptotic cases of very large, or very small wavelengths. Finally, it is also of interest to compare equation [7.161] with that of the piston-fluid system provided with anechoic terminal impedance [7.107], which is repeated here for convenience as: κ 02 − κ 2 + i μ f κ = 0 κ=
ω L ωL ; κ 0 = 0 ; ω0 = cf cf
M K0 ; μf = f ; M f = ρf Sf L M0 M0
[7.163]
As a major difference, in this particular case the piston-fluid system is found to reduce to a standard damped harmonic oscillator in which vibroacoustic coupling force reduces to damping whereas the sphere does not. The reason for such a difference lies in the presence of two distinct components in the spherical pressure field instead of only one in the case of plane waves, as already pointed out in Chapter 5 subsection 5.3.4.1. The complex amplitude of the fluctuating velocity u of the fluid particles is obtained from the (linearized) radial momentum equation: iω u = −
1 dp ρ f dr
[7.164]
With the aid of the pressure field [7.148], one obtains: u=
−iω R02U 0 (1 − ikR0 ) (1 + ik r ) iωθ e 2 r2 1 + ( kR0 )
[7.165]
Contrasting with the case of plane waves, formulas [7.148] and [7.165] show that in spherical waves, the shapes of the pressure field and that of the particle velocity field behave differently in relation to the distance r from the source. As already noted earlier, the pressure field decreases as 1/r while the velocity field is characterized by a near field component which prevails in the range kr << 1 where it decreases essentially as ( R0 / r ) and a far field component which decreases as 1/r. 2
Of course, the near field component corresponds to an incompressible motion as it can be checked by comparing the present results to those established in Chapter 2, subsection 2.2.3.1. As indicated earlier, the added mass is not infinite, precisely because the near field component decreases sufficiently fast to let the kinetic energy of the fluid be finite. Such a near field component does not exist in the case of plane waves, which explains why in the absence of wave reflection, there is no added mass on the piston. Returning to the spherical waves, it can be stated that the far field component behaves in the same manner as plane waves, since at any place in the far field the pressure to velocity ratio is found to be equal to the specific impedance of the medium.
636
Fluid-structure interaction
Figure 7.33. Radial profile of the velocity field, full line: total field, dashed line: far fied
In Figure 7.33, two plots of the radial profile of the fluid velocity are presented, radial distance r is scaled by using the wavelength λ. In each plot, the total field (full line) and the far field component (dashed line) are superposed to point out the differences which are essentially limited to a typical distance of about one wavelength from the sphere surface ( R0 = 0.5 m ). Computation is performed in air at STP assuming a vibration amplitude U 0 = 0.1 mm . Maximum pressure at the surface of the sphere has been reported on the graphs. As an instructive exercise, radiation damping can be related to the radiated energy. To calculate the sound intensity, the real parts of the complex fields [7.148] and [7.165] are retained. One obtains: ω 2 ρ f R02U 0 ( cos ωθ + k R0 sin ωθ )
Re ( p ) =
(1 + ( k R ) ) r 2
[7.166]
0
Re ( u ) =
−ω R02U 0 ( (1 + kR0 ) sin ωθ + k ( r − R0 ) cos ωθ )
(1 + ( kR ) ) r 2
0
2
[7.167]
Although, the near field component could be safely discarded to alleviate the mathematical manipulations, it is retained in [7.167] as part of the exercise. The instantaneous sound intensity radiated through a concentric spherical surface of arbitray radius r larger than or equal to R0 is expressed as: I ( r; t ) = Re( p ) × Re(u ) =
( k ( r − R ) cos 0
2
ω 3 ρ 0 R04U 02 1 × 2 2 3 1 + ( kR ) r
(
0
)
ωθ + kR0 (1 + kR0 ) sin 2 ωθ + 0.5 ( kR0 ) k ( r − R0 ) sin 2ωθ )
The mean intensity radiated per cycle is:
[7.168]
Energy dissipation by the fluid
ω I (r ) = 2π
⌠ t1 ⎮ ⎮ ⎮ ⌡t
2π
+
ω
I ( r ) dt =
{
ω 3 ρ f R04U 02 k r + ( kR0 )
1
(
2 1 + ( kR0 )
)r
2 2
2
}
3
637
[7.169]
At this step, only the main far field term whithin the braces is retained to produce the mean sound power: PR ( r ) = 4π r 2 I ( r ) =
2πω 4 ρ f R04U 02
(
c f 1 + ( kR0 )
)
2 2
[7.170]
A result which can be expressed in the compacted form: PR =
ω 3kM aU 02
(
2 1 + ( kR0 )
[7.171]
)
2 2
In the range of large wavelengths kR0 << 1 , formula [7.171] is further simplified to give: PR =
ω 3kM aU 02 = ω kE 2
[7.172]
where E can be interpreted as a characteristic kinetic energy closely related to the added mass effect. Radiated energy follows as: ER =
2π M ω 2U 02 kR0 PR = π a 2 2 ω 1 + ( kR0 )
(
)
[7.173]
ER is to be compared with that energy which would be dissipated by the oscillator according to the viscous damping model [7.158]. With the aid of relation [7.7], the dissipated energy is written as: Ed = 4πζ 1Em
[7.174]
where the mechanical energy of the oscillator is Em = ( M 0 + M a ) ω 2U 02 / 2 . From the equivalent viscous damping formula [7.157], one obtains: Ed = π M aω 2 k1 R0U 02
[7.175]
Comparing the approximate result [7.175] to the exact result [7.173], the same conclusions may be drawn as earlier, that the equivalent viscous model largely overestimates the actual energy loss in the range of large wave numbers kR0 ≥ 1 . Finally, Figure 7.34 illustrates the variation of the radiation efficiency of the sphere in relation with the dimensionless wave number. The logarithmic plot refers to the same spherical shell as that used for the above illustrative calculations. It can be noted that the slope of the curve changes below and above the characteristic value
638
Fluid-structure interaction
kR0 = 1 which separates the domain dominated by inertial effect and that dominated by sound radiation.
Figure 7.34. Sound power radiated by a spherical shell pulsating according to the breathing mode, versus the dimensionless wave number
7.2.1.6 Kirchhoff-Helmholtz integral applied to the spherical radiator The method used in the last subsection to solve the problem of vibroacoustic coupling and sound radiation for a pulsating spherical shell is straightforward and corresponds to the standard analytical technique to solve fluid-structure coupled systems, as already illustrated through many examples in the present book. Nevertheless, the example of sound radiation in a semi-infinite space by a vibrating baffled piston also illustrated the interest to shift from a local and differential formulation of the problem to that of a surface integral formulation of it. It turns out that the problem of the spherical radiator gives us a good opportunity to clarify certains aspects of the integral formulation, as emphasized in [TEM 01], which was of great help in the writing of this subsection. As a first attempt to solve the problem, the same method which was successfully used to analyse the baffled piston is applied to the present problem. Hence the radiated pressure field at some vector position r is formulated as the Rayleigh integral: ρ p( r ; t ) = − f 4π
⌠ (e) ⎮ Q ⎮ ⎮ ⎮ ⌡(S )
( r ; t − r / c ) U ( t − r / c )dS 0
f
r
f
[7.176]
where r = r − r0 is the distance between any generic point P(r ) in the fluid and any generic source location P0 (r0 ) on the vibrating surface, as sketched in
Energy dissipation by the fluid
639
Figure 7.35. Notice that, in contrast with the baffled piston, the waves radiated at location P0 will reach every point in space, even those located “behind” the plane tangent to the sphere at P0 .
P0 r0 ϕ
P
r
α r
Figure 7.35. Sound radiated by a pulsating sphere: contribution of a point source at P0
The generic distance r can obviously be obtained from: R0 sin ϕ = r sin α ; R0 cos ϕ + r cos α = r
[7.177]
Hence: r = r 2 + R02 - 2rR0 cosϕ
[7.178]
Figure 7.36. Two emissive point sources located either on the same hemisphere, or not
Notice that the same formulation [7.178] for r applies, either when P(r ) and P0 (r0 ) are on the same hemisphere or when they lie on opposite hemispheres, see Figure 7.36. On the other hand, the elementary source at the sphere surface is given by:
640
Fluid-structure interaction
iω t − kr Q ( e ) ( r; t − kr ) = −ω 2U 0 e ( )
[7.179]
The Rayleigh integral then becomes: ⌠
π
2 ⎮ 0 ⎮
2
ω ρ f U0 R p( r ; t ) = 2
⎮ ⎮ ⎮ ⎮ ⌡0
1 r 2 + R02 − 2rR0 cos ϕ
e
⎛ r 2 + R02 − 2 rR0 cosϕ iω ⎜ t − ⎜ cf ⎝
⎞ ⎟ ⎟ ⎠
sin ϕ dϕ
[7.180]
which can be expressed as: p(r;t) =
2
2
ω ρ f U 0 R0 1 i (ωt − kr ) e 2ikR0 r
(
)
e−ikR0 − e+ikR0 =
−ω ρ f U 0 R0 sin( kR0 ) ei (ωt − kr ) 2
2
kR0
r
[7.181]
By comparing the proposed solution [7.181] to the exact solution [7.148], it may be concluded that some important point is missed by applying the first Rayleigh integral to solve the present problem. In order to understand where the missing term is lying, we must refer to the Kirchhoff-Helmholtz integral [5.299]. In the present problem, due to the central symmetry it takes the following convenient form: p( r; ω ) = p( R0 ; ω ) I1 ( r ) − ω 2 ρ f U 0 I 2 ( r ) ∀r > R0
[7.182]
where I1 ( r ) and I 2 ( r ) are the two surface integrals associated to the pressure source term and the volume velocity source term, respectively. In fact, the part of the solution arising from the volume velocity source term has been determined already by solving the first Rayleigh integral, see formula [7.181]. I 2 ( r ) was found to be: I2 ( r ) =
R02 sin( kR0 ) kR0 r
[7.183]
Before embarking in the calculation of the first term, some qualitative comments about its nature are in order. First, we note that in [7.182] it stands for a dipole source, as explained in Chapter 5, subsection 5.3.5.5. One may rightly wonder why a dipole source term must be considered in the present problem since the pulsating sphere clearly behaves as a monopole source solely. Going a step further, if we accept the idea, it can be questioned why a dipole source is to be retained in the integral formulation of the present problem, and not in the case of the baffled piston. At this respect, some clue can be obtained by comparing Figures 7.34 and 7.35. Indeed, considering the equatorial plane which separates the unshaded and shaded hemispheres, both of them contribute to the pressure field observed at P. This would also hold for the baffled piston, provided it is replaced by a very thin pillbox (thickness ε in Figure 7.37) each face vibrating according to the same harmonic motion of opposite complex amplitude, in the Oy direction of unit vector j .
Energy dissipation by the fluid
641
Figure 7.37. Vibrating unbaffled piston modelled as a thin pillbox
Clearly, the pressure field at any external position P can be obtained by adding the pressure waves excited at both of the pill box faces. Furthermore, due to the symmetry of the problem with respect to the midplane y = 0 and since ε is supposed to be infinitesimal, the pressure field at P is twice that value which would be induced by the motion of one face only. This is precisely the reasoning we followed in Chapter 5, where it was presented as an application of the image source method. The major difference with the present problem is that due to the finite radius of the sphere, when summing the contributions of a pair of emissive points located symmetrically with respect to the equatorial plane, it is necessary to account for the retarded time which varies with the angle ϕ . It remains now to confirm the idea by proceeding to a quantitative analysis. Though the pressure on the sphere is not known a priori, it can be easily determined by using [7.182] in the immediate vicinity of R0 which gives: p( R0 ; ω ) = −ω 2 ρ f U 0
I1 ( R0 ) 1 + I 2 ( R0 )
[7.184]
Accordingly, the resulting field is found to be proportional to the velocity of the pulsating sphere as should be expected: ⎛ I ( r ) I1 ( R0 ) ⎞ p( r; ω ) = −ω 2 ρ f U 0 ⎜⎜ I 2 ( r ) + 1 ⎟ ∀r > R0 1 + I 2 ( R0 ) ⎟⎠ ⎝
[7.185]
The remaining task needed to determine entirely the radiated pressure field is to calculate the surface integral: 1 I1 ( r ) = 4π
⌠ ⎮ ⎛ ⎡ 1 ⎮ grad ⎢ ⎮ ⎜ ⎣r ⎮ ⎝ ⌡( S )
⎤⎞ e − ikr ⎥ ⎟ .n ( r0 )d S ⎦⎠
[7.186]
642
Fluid-structure interaction
It is found convenient to transform [7.186] into a volume integral taken in the sphere volume (V ) : I1 ( r ) =
1 4π
⌠ ⎮ ⎡1 ⎮ Δ⎢ ⎮ ⎣r ⎮ ⌡(V )
⎤ e − ikr ⎥ dV ⎦
[7.187]
Remembering that the argument of the Laplacian operator in [7.187] is nothing else than the Fourier transform of the Green function (see equation [5.197]) and that r never vanishes within (V ) , by definition at any point outside (V ) , the following homogeneous wave equation is verified: 2
⎡1 ⎤ k Δ ⎢ e − ikr ⎥ + e − ikr = 0 ⎣r ⎦ r
[7.188]
Therefore [7.187] can be transformed into: k2 I1 ( r ) = − 4π
⌠ ⎮ ⎮ ⎮ ⎮ ⌡(V )
1 − ikr e dV r
[7.189]
Incidentally, the form [7.189] of I1 indicates that the associated pressure component
represents the contribution of a monopole source distributed uniformly within (V ) . I1 can be solved analytically rather easily. It is first written as: k2 I1 ( r ) = − 2
R
π
⌠ 0 ⎮ r 2 dr 0 0 ⎮ ⌡0
⌠ ⎮ ⎮ ⎮ ⎮ ⌡0
1 − ikr e sin ϕ dϕ r
[7.190]
where r is given by [7.178], which suggests transforming the integration variable ϕ into r in the angular integral at constant r0 . The particularly simple result is immediately obtained: R
r+r ⌠ 0 k 2 ⌠ R0 k 2 e − ikr I1 ( r ) = ⎮⎮ r0 dr0 ⎮⎮ e − ikr dr = ⌡r − r0 r 2r ⌡0
⌠ 0 ⎮ ⎮ ⎮ ⎮ ⌡0
(e
+ ikr0
− e − ikr0 )
2ik
r0 dr0
[7.191]
and finally, I1 ( r ) =
e − ikr kr
R
⌠ 0 ⎮ sin ⎮ ⌡0
( kr0 ) kr0 d ( kr0 ) =
e − ikr ( sin kR0 − kR0 cos kR0 ) kr
[7.192]
Substituting the coefficients [7.182] and [7.192] into [7.185] and carrying out a few trigonometric manipulations, the final result can be put in the form [7.148]. Finally, if the same analytical procedure is applied to the pillbox, it may be verified that the
Energy dissipation by the fluid
643
contribution of the volume term vanishes, which justifies the use of the first Rayleigh formula to solve the problem of the baffled piston. 7.2.1.7 Rigid sphere oscillating rectilinearly
Figure 7.38. Rigid sphere in harmonic oscillation along the Oz direction
The problem we are interested in is sketched in Figure 7.38 and formulated as the following differential system: 2
∂ ⎛ 2 ∂p⎞ 1 ∂ p ∂ 2 p ⎛ ωr ⎞ + +⎜ ⎟ p=0 ⎜r ⎟+ ∂ r ⎝ ∂ r ⎠ tan ϕ ∂ϕ ∂ϕ 2 ⎝⎜ c f ⎠⎟ ∂p = −iωρ f U s eiω t = ω 2 ρ f Z 0 cos ϕ eiω t ∂ r r=R
[7.193]
0
where U s is the radial component of the sphere velocity oscillating vertically with amplitude Z 0 . Furthermore, the fluid is supposed to extend to infinity and the solution of interest is an outgoing wave. From the qualitative standpoint, the major difference between the present problem and those presented above, is that here the solid vibrates without changing its volume. Accordingly, the nature of the associated acoustic source is changed from a volume velocity or monopole source, to that of a pressure, or dipole source. Solving equations [7.193] presents no particular difficulty. The pressure field is supposed to be of the separated form: p( r, ϕ , θ ; ω ) = A( r ) cos ϕ e (
i ω t − kr )
[7.194]
644
Fluid-structure interaction
Since only the outgoing wave is to be retained, the radial profile A( r ) is a spherical Hankel function of first order, instead of a spherical Bessel function, as it would be found for standing waves (see [5.97]). i ⎞ − ik ( r − R0 ) ⎛ 1 A ( r ) = Ph 1 1 ( kr ) = P1 ⎜ 2 2 + ⎟e kr ⎠ ⎝k r
[7.195]
The field is then suitably matched to comply with the fluid-structure interface condition. One obtains: P1 =
ρ f c 2f k 4 R03 Z 0 cos ϕ
[7.196]
k 2 R02 − 2 (1 + ikR0 )
Whence the radiated pressure field: p+ =
ρ f ω 2 R03 Z 0 cos ϕ ( k 2 R02 − 2 + 2ikR0 ) 1 + ikr
( kR0 )
4
+4
r2
e
− ik ( r − R0 )
[7.197]
The dipolar nature of the pressure field can be asserted by comparing [7.197] to [5.221]. Comparison can be further continued by letting kR0 tend to zero and retaining the far field component only. By identification with [5.222] the strength of the dipole equivalent to the osillating sphere is found to be: M ( ) = ( 2π R02iω Z 0 ) ( 4 R0 ) e
[7.198]
In [7.198], the term between the first pair of parentheses stands for the volume velocity, or volume flow rate, of the fluid displaced by the sphere osillation and the term between the second pair of parentheses is interpreted as the lever arm of the dipole. Finally, it is left to the reader to verify that the near field component can be interpreted as the inertial component already described in Chapter 2, subsection 2.3.5.3. Turning bak to the far field, the radial velocity of the fluid particles Uu is obtained through the radial momentum equation as: U=
2 2 2 ∂p+ iω Z 0 R0 ( k R0 − 2 + 2ikR0 ) ( kR0 ) cos ϕ − ik ( r − R0 ) = e 4 ρ f ω ∂r r2 ( kR0 ) + 4
i
[7.199]
Sound intensity follows as: 2 4 1 1 ρ f c f (ω Z 0 ) ( kR0 ) ⎛ R0 cos ϕ ⎞ I ( r ; t ) = Re ( p+U * ) u = ⎜ ⎟u 4 2 2 r ⎝ ⎠ ( kR0 ) + 4
[7.200]
It varies with the colatitude angle according to a cos2 ϕ law, leading to a figureeight directivity pattern which is a basic characteristic of sound radiated by a dipole source, see Figure 7.39, where it is represented as linear and logarithmic polar plots. The radiated power is given by:
Energy dissipation by the fluid
2π R02 ρ f c f (ω Z 0 ) ( kR0 ) 4 2 ( kR0 ) + 4 2
PR =
4
π
⌠ ⎮ ⎮ ⌡0
4π R02 ρ f c f (ω Z 0 ) ( kR0 ) 2
cos2 ϕ sin ϕ dϕ =
(
6 4 + ( kR0 )
4
645
4
)
[7.201]
Figure 7.39. Directivity diagram of dipole radiation
646
Fluid-structure interaction
Figure 7.40. Power radiated by a rigid sphere. Dashed line: breathing mode (monopole source), full line: rectilinear rigid mode (dipole source)
Figure 7.40 illustrates the variation of the radiation efficiency of the sphere with the dimensionless wave number. Dashed line reproduces the plot of Figure 7.34 to help comparison between monopole (dashed line) and dipole radiation (full line). In both cases, the slope of the curve changes below and above a characteristic value kR0 1 which separates the domain dominated by inertial effect and that dominated by sound radiation. It is also noted that in the range of low wave numbers, dipole radiation is much less efficient than monopole radiation, though it increases faster with kR0 . In the range kR0 > 1 the curves become very similar to each other. Finally, it is of interest to evaluate radiation damping related to the rectilinear rigid mode. For this purpose equation [7.7] is found to be particularly convenient. The 1 2 mechanical energy of the oscillator is Em = M e (ω Z 0 ) while that dissipated per 2 cycle by radiation is ER = ( 2π / ω ) PR . With the aid of [7.201], the energy loss factor can be expressed as:
( kR0 ) ER M R = = 4πζ Em M e 6 4 + ( kR0 )4 4
(
)
[7.202]
where M e is the mass of the sphere plus the fluid added mass, which in this case, is half the physical mass of displaced fluid (see relation [2.245]). M R is a mass coefficient related to sound radiation which is found to be: M R = 4π R02 λρ f
[7.203]
Energy dissipation by the fluid
647
Here λ designates the wavelength of the emitted sound. Considering a spherical thin shell of thikness h, the relation [7.202] can be expressed as: μ ( kR0 ) ER =ζ = R 4 4π Em 4 + ( kR0 ) 3
(
)
;
μR =
1 ⎛ 6h ρ S 2 ⎜1 + ⎜ Rρ 0 f ⎝
⎞ ⎟⎟ ⎠
[7.204]
Figure 7.41. Radiation damping of the sphere for a rectilinear rigid mode
The damping ratio defined by relation [7.204] has a marked maximum which depends on the nondimensional mass parameter μR as illustrated in Figure 7.41, where a spherical aluminium shell of radius R0 = 0.5 m and thickness h = 1 mm is assumed to vibrate in air at STP (left-hand plot) and in water (right-hand plot). Presence of a maximum does not mean that radiated energy itself has a maximum but merely that when kR0 is increased from very small values up to the value of maximum damping ratio, the amount of radiated energy increases faster than the mechanical energy of the radiator and when it is further increased the reverse occurs. On the other hand, damping is found to be much higher in water than in air due to the large change in the mass parameter. Nevertheless, it is necessary to emphasize that practical validity of the damping values presented here can be questioned for several reasons; in particular due to the presence of other dissipative mechanisms which have to be accounted for when radiation damping is small and for practical difficulties in achieving such experiments in water to obtain high damping values. 7.2.1.8 Radiation of circular cylindrical shells Let us consider a circular cylindrical shell of radius R0 , which is assumed to be either of infinite length, or at least sufficiently elongated that a strip model can be used for mathematical convenience, see Figure 7.42. The radial displacement is of the type: U n cos nθ e
iω t
[7.205]
648
Fluid-structure interaction
Figure 7.42. Radiation of cylindrical sound waves
The cylinder is assumed to be immersed in an infinite extent of a homogeneous fluid. The vibroacoustic system is formulated by using the same strip model as that introduced earlier in Chapter 2 subsection 2.3.1. Concentrating on the fluid, we have to solve the following boundary value problem: ∂ 2p 1∂ p 1 ∂ 2p + + + k2 p = 0 ∂ r2 r ∂ r r2 ∂ θ 2 ∂p ∂r
2
[7.206]
2 f
= k ρ f c U n cos nθ Ro
where again k is the wave number and U n is the amplitude of the structural mode of vibration. Here also the solution of interest is an outgoing wave. Using the results established in Chapter 5, subsection 5.1.1.2, the general solution is first written as: pn ( r, θ ; t ) = { AJ n ( k r ) + BYn ( k r )}cos nθ eiω t ∀r ≥ R0
[7.207]
Since we have to deal here with radially travelling waves, it is preferred to rewrite [7.207] in terms of Hankel functions instead of Bessel functions. They are defined as: H
(1) n
( z ) = J n ( z ) + iYn ( z )
;
H
(2) n
( z ) = J n ( z ) − iYn ( z )
[7.208]
Energy dissipation by the fluid
649
With the aid of [7.208], the field [7.207] is rewritten in the intermediate form: pn ( r, θ ; t ) =
{( A − iB ) H ( ) ( k r ) + ( A + iB ) H ( ) ( k r )}cos nθ e 1 n
2
n
iω t
∀r ≥ R0
[7.209]
Concentrating on the far field component kr >> 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 , using the first term of their series expansions, the Bessel functions can be suitably approximated as:
650
Fluid-structure interaction
Jn ≅
1 ⎛ kR0 ⎞n ⎜ ⎟ n! ⎝ 2 ⎠
kR 2 Y0 ≅ Log 0 π 2
; Yn ≅ −
( n − 1)! ⎛
2 ⎞ ⎜ ⎟ ⎝ kR0 ⎠
π
n
4 ; Y0′ ≅ π kR0
[7.215]
After a few straightforward manipulations the inertia force [2.148] is recovered and the following dissipative force is found: Im ( Qn ) =
ω2Ma
⎛ kR ⎞ 2π ⎜ 0 ⎟ 2 ( ( n − 1)!) ⎝ 2 ⎠
2n
[7.216]
Whence the modal damping ratio: ςn =
Ma Ms + Ma
2π
⎛ kR0 ⎞ ⎟ 2 ⎜ ( ( n − 1)!) ⎝ 2 ⎠
2n
[7.217]
Computation of the radiated power follows the same procedure as in the case of the sphere. Depending on the circumferential order of the mode, the source is of the monopole (n = 0), dipole (n = 1) or multipole (n >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
Fluid-structure interaction
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
Energy dissipation by the fluid
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
Fluid-structure interaction
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
Energy dissipation by the fluid
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
Fluid-structure interaction
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
Energy dissipation by the fluid
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
Fluid-structure interaction
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]
Energy dissipation by the fluid
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
Fluid-structure interaction
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
Energy dissipation by the fluid
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
Fluid-structure interaction
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:
Energy dissipation by the fluid
⎛ ⎛ 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
Fluid-structure interaction
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.
Energy dissipation by the fluid
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
Fluid-structure interaction
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]
Energy dissipation by the fluid
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
Fluid-structure interaction
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
Energy dissipation by the fluid
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
Fluid-structure interaction
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]
Energy dissipation by the fluid
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
Fluid-structure interaction
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:
Energy dissipation by the fluid
⌠
⌠ ⎮ ⎛ ⎛ 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
Fluid-structure interaction
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]
Energy dissipation by the fluid
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
Fluid-structure interaction
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.
Energy dissipation by the fluid
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
Fluid-structure interaction
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]
Energy dissipation by the fluid
⎛ ⎛ 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
Fluid-structure interaction
⎡ ⎤ 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]
Energy dissipation by the fluid
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 << 1 When the tube radius is so small that viscous forces prevail on the inertia forces in most part of the tube cross-section at least, the radial profile of fluid velocity tends to a parabolic shape, as illustrated in Figure 7.61. Damping ratio is of course much larger than in the poorly confined case. It is given by the asymptotic formula: ςν
νf 3M a′ M s + M a′ 2ω1 R 2
; M a′ ≅
6 Mf 5
[7.304]
Notice that the added mass M a′ coefficient is larger by about 20% than the inviscid value M a = M f .
Figure 7.61. Radial variation of the axial fluid velocity at distinct times expressed as a fraction of the period of oscillation (small value of Stokes number)
682
Fluid-structure interaction
7.3.2.2 Flexible plates coupled by a liquid layer The importance of viscous forces in fluid-structure coupling, in relation to the fluid confinement ratio, which has been put in evidence in the case of a circular cylindrical tube, is qualitatively the same whatever the detailed geometry may be. The basic reason is that in a highly confined fluid, viscous shearing is essentially governed by the velocity derivatives taken along the direction of confinement. It is of interest to illustrate this point taking another example which can be easily solved analytically and which provides us at the same time with a pertinent idealization of the thin films of fluid, those used in lubrification for instance. The system is sketched in Figure 7.62. It is composed of two identical rectangular plates P1 and P2 which are parallel and coupled to a layer of incompressible and viscous liquid. Calculation is alleviated by assuming that the width Lx is much smaller than the length Lz . Furthermore, the liquid layer thickness 2h is assumed to be much smaller than Lx . The plates are hinged along the Oz direction, and vibrate according to the bending mode shapes:
b g
b g
Yn( ) ( x ) = U n( ) sin kn x ; 1
kn =
1
nπ Lx
; λn =
Yn(
2)
( x ) = U n( 2) sin kn x
Lx >> 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:
Energy dissipation by the fluid
⎧ ⎪ ⎪ ⎪ ⎪⎪ ⎨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
Fluid-structure interaction
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]
Energy dissipation by the fluid
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
Fluid-structure interaction
( 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 and 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]
Energy dissipation by the fluid
ς n( ) =
M n(
ν
(
a)
(s)
νf (a )
2 Mn + Mn
)
2ωn h 2
<< 1
687
[7.327]
In the highly confined range Sν < 1, it takes on the form: ς n( ) =
3.6 M n(
ν
(
a)
(s)
νf (a)
2 M n + 1.2 M n
)
[7.328]
ωn h 2
According to formula [7.328], the mode in phase opposition is likely to be overdamped. To conclude on this exercise, it is also interesting to relate directly the forces exerted by the fluid to the longitudinal fluctuating velocity field vn ( y ) . Relation [7.320] shows that the generalized force is proportional to the fluctuating pressure, which in turn is given by relation [7.317]. On the other hand, vn ( y ) is given by relation [7.314]. Then it is not difficult to show that: d 2 vn dy 2
= αn k 2 = y =± h
ρν kn p ⇒p= f f kn ρ fν f
⎛ d 2v ⎜ 2n ⎜ dy ⎝
⎞ ⎟ ⎟ y =± h ⎠
[7.329]
Form [7.329] is especially well suited to clearly demonstrate the relationship between the generalized force and the curvature of the transverse velocity profile at the wall. Therefore, in contrast with the kinetic energy which is a volume integral property of the fluid in motion, dissipation by viscosity is an integral property concentrated at the fluid-structure interface. As a consequence, assessing viscous dissipation is much more demanding in computational effort than added mass. Figure 7.64 shows a few plots of the longitudinal and transverse velocity components at several times during half a cyle of vibration, after normalisation to the maximum of the plate velocity during the cycle. The upper plots refer to a high Stokes number Sν = 103 . As expected, the velocity field is largely dominated by the inertial component except in the thin boundary layer at the walls. The lower plots are for an intermediate value Sν = 30. Viscous effects are noticeable over the whole fluid layer. If Sν is further decreased, vn tends asymptotically to a y 2 law profile, whereas un tends to vary as y. 7.3.2.3 Rigid plate coupled to a thin liquid layer Geometry of the problem is shown in Figure 7.65. It can idealize a gap filed with a liquid which separates a structure and a support. Both of them can be assumed to be rigid, at least at the local scale of the supported zone. However, to alleviate calculation, length of the system in the Oy direction is assumed to be sufficiently large to validate the strip model presented here.
688
Fluid-structure interaction
Figure 7.64. Profiles of the longitudinal and transverse fluid velocities at a few times during half a cycle of vibration
Figure 7.65. Geometry of the thin fluid layer squeezed by a rigid plate
Let Z denote the vibration amplitude of the plate, which is still assumed here to remain much smaller than thickness h of the fluid layer in order to legitimate a linear model. On the other hand the confinement ratio L/h is assumed to be very large. Accordingly, the thin fluid layer approximation is also valid. Hence, fluid velocity is approximated by the purely longitudinal field V ( x, z; ω ) = v ( x, z; ω )i and z
Energy dissipation by the fluid
689
dependence of pressure is discarded. As in the inviscid case, the mass equation is integrated in the elementary volume hdx, to relate the mean fluid velocity to that of the solid, which gives: h
∂v Z dz = 0 ⇒ v ( x; ω ) = iω x ∂ x h ⌡0 ⌠
−iω Z + ⎮⎮
[7.330]
where v denotes the mean longitudinal velocity of the fluid, as averagd over the layer thickness. In the same way, the longitudinal moment equation is also integrated with respect to z. With the aid of [7.330] one obtains: ∂ v ⎫⎪ dp Z ρ ν ⎧⎪ ∂ v = ω2ρ f x + f f ⎨ − ⎬ dx h h ⎩⎪ ∂ z h ∂ z 0 ⎭⎪
[7.331]
As could have been anticipated, to determine the mean fluctuating pressure field p , taking into account viscosity, we need to know the transverse profile of the fluctuating longitudinal velocity, more specifically the tranverse component of the gradient at the wall. Thus the problem is significantly more complicated than in the inviscid case. Nevertheless, the preceding examples indicated that, provided Sν is sufficiently small, the velocity profile can be reasonably well approximated by using a parabolic profile. Adopting such a simplifying assumption, the z dependence of the local field v ( x, z; ω ) can be entirely determined by using the boundary conditions at the walls and by adjusting the remaining constant to the condition of conservation fluid material. Final result is: v ( x, z; ω ) = iω
6Z xz ( h − z ) h3
[7.332]
Substituting the field [7.332] into equation [7.331], the mean pressure field is finally obtained as: p ( x; ω ) = ω 2 ρ f
Z ⎛ 12iν f ⎞ 2 2 ⎜1 − ⎟(x − L ) 2h ⎝ ωh2 ⎠
[7.333]
The generalized force exerted on the vibrating plate follows as: Q = ω2ρ f
12ν f 2 L2 L ⎛ ⎜1 − i ωh2 3 h⎝
⎞ ⎟Z ⎠
[7.334]
Within the approximations of the model, the added mass (per unit length in the Oy direction) is found to be the same as in inviscid fluid: Ma = ρ f
L2 L 12 h
[7.335]
690
Fluid-structure interaction
The equivalent viscous damping ratio is found to be: ςν =
12 M a M a 12ν f = Sν M s + M a ωh2 M s + M a
[7.336]
As a consequence of the parabolic profile, damping is found to be proportional to the Stokes number. 7.3.2.4 Cylindrical annular gap The example considered here concerns the fluid-structure interaction which occurs at an annular gap filled with a liquid between a vibrating tube and a support plate, see Figure 7.66. The radius of the circular cylindrical tube is denoted R, H denotes the thickness of the plate and R+h the bore radius, where h is the gap between the tube and the plate, assumed to be uniform for the sake of mathematical simplicity. We are mainly interested in evaluating the amount of damping provided by viscous shearing of the fluid within the gap when the tube vibrates according to a bending mode with small amplitude X 0 << h . Such a problem is of practical interest to perform design analyses concerning the risks of excessive flow-induced vibration in tube and shell heat exchangers as will be detailed in Volume 4 of the present book series. The analysis presented here is largly inspired from that found in [CHE 81], which, to the author’s knowledge, was the first published in the open literature on the subject.
Figure 7.66. Tube loosely supported at a plate provided with a circular cylindrical bore. As the bore radius is larger than the tube radius, there is a fluid annular gap between the plate and the tube
Energy dissipation by the fluid
691
The problem is first formulated by using cylindrical coordinates as indicated in the figure. The fluid velocity field is written as: V = uu + vu1 + wk [7.337] The Navier-Stokes equations follow as: ∂u u 1 ∂v ∂ w + + + =0 ∂ r r r ∂θ ∂ z iωu +
⎛ ∂2 u 1 ∂u u 1 ∂2 u 2 ∂v ∂2 u ⎞ 1 ∂p −ν f ⎜ 2 + − + − + ⎟=0 r ∂ r r 2 r 2 ∂θ 2 r 2 ∂θ ∂ z 2 ⎠ ρ f ∂r ⎝ ∂r
1 ∂p iω v + −ν f ρ f r ∂θ iω w +
⎛ ∂2 v 1 ∂v v 1 ∂2 v 2 ∂u ∂2 v ⎞ − + − + ⎜ 2+ ⎟=0 r ∂ r r 2 r 2 ∂ θ 2 r 2 ∂θ ∂ z 2 ⎠ ⎝ ∂r
[7.338]
⎛ ∂2 w 1 ∂ w 1 ∂2 w ∂2 w ⎞ 1 ∂p −ν f ⎜ 2 + + + ⎟=0 r ∂ r r 2 ∂θ 2 ∂ z 2 ⎠ ρ f ∂z ⎝ ∂r
The domain of interest is R ≤ r ≤ R + h , 0 ≤ θ ≤ 2π , 0 ≤ z ≤ H . The conditions to be fullfiled by the fluid at the boundaries of the annular gap are: ur = R + h = vr = R + h = wr = R + h = 0 0 ≤ z ≤ H ⎧ ⎪ ⎨ur = R = iω X 0 cos θ ; vr = R = −iω X 0 sin θ 0 ≤ z ≤ H ⎪ wr = R = 0 0 ≤ z ≤ H ; p z = 0 = pz = H = 0 ⎩
[7.339]
However, the general form of equations [7.338] can be drastically simplified if the gap thickness is sufficiently small, as is often the case in practice. In fact, if h is much smaller than R, one can check that the curvature terms are negligible. Therefore, the problem turns out to be similar to that of the rigid plate presented in the last subsection. Formal proof is given as follows. First, to determine the relative magnitude of the different terms in the Navier-Stokes equations it is appropriate to rewrite [7.338] into a nondimensional form by using the following reduced quantities: r−R u v ; u= ; v= ωh ωR h ph z p= 2 z= ; ω ρ f R2 X 0 H
r=
;
w=
w ωH
[7.340]
Notice that here the upper bars are used to make the distinction between the nondimensional and the dimensional quantities and not for denoting a space averaging process. The dimensionless Navier-Stokes equations are found to be:
692
Fluid-structure interaction
∂ u hu ∂ v ∂ w + + + =0 ∂ r R ∂θ ∂ z 2 2 2 2 ∂ 2 u ⎞ 2h ∂ v ⎛ h ⎞ ∂ 2 u ⎛ R ⎞ ⎛ X ⎞ ∂ p ν f ⎛ ∂ u h ∂u ⎛ h ⎞ ⎞ ⎛ − 2⎜ 2+ − ⎜ ⎟ ⎟⎜u − 2 ⎟ − +⎜ ⎟ =0 iu + ⎜ ⎟ ⎜ 0 ⎟ ∂θ ⎠ R ∂θ ⎝ H ⎠ ∂ z 2 R ∂ r ⎝ R ⎠ ⎟⎠ ⎝ ⎝ h ⎠ ⎝ h ⎠ ∂ r ω h ⎜⎝ ∂ r 2 2 2 ∂ 2 v ⎞ 2h 3 ∂ u ⎛ h ⎞ ∂ 2 v ⎛ X ⎞ ∂ p ν f ⎛ ∂ v h ∂v ⎛ h ⎞ ⎞ ⎛ − 2⎜ 2+ − ⎜ ⎟ ⎟⎜v − 2 ⎟ + 3 +⎜ ⎟ iv + ⎜ 0 ⎟ =0 ∂θ ⎠ R ∂θ ⎝ H ⎠ ∂ z 2 R ∂ r ⎝ R ⎠ ⎟⎠ ⎝ ⎝ h ⎠ ∂θ ω h ⎜⎝ ∂ r 2 2 2 2 2 2 ⎛ R ⎞ ⎛ X ⎞ ∂ p ν f ⎛ ∂ w h ∂w ⎛ h ⎞ ∂ w ⎛ h ⎞ ∂ w ⎞ iw + ⎜ ⎟ ⎜ 0 ⎟ − 2⎜ 2+ +⎜ ⎟ + ⎟=0 ⎜ ⎟ R ∂r ⎝ R ⎠ ∂θ 2 ⎝ H ⎠ ∂ z 2 ⎟⎠ ⎝ H ⎠ ⎝ h ⎠ ∂ z ω h ⎜⎝ ∂ r [7.341]
Major interest of formulation [7.341] is to make conspicuous already at first glance the dimensionless parameters which are physically pertinent to discuss the problem, namely the reciprocals of the radial R / h and axial R / H confinement ratios and the reciprocal of the Stokes number ω h 2 / ν f referred to the gap width. In the case of tube heat exchangers, h is much smaller than R while H is often of the same order of magnitude as R. In such configurations, equations [7.341] can be simplified as: ∂u ∂ v ∂ w ⎧ + + =0 ⎪ ∂ r ∂θ ∂ z ⎪ ∂p ⎪ =0 ⎪ ∂r ⎪ ⎨ 2 ⎛ X ⎞∂p νf ⎛∂ v ⎞ ⎪ − 2 ⎜ 2 ⎟=0 iv + ⎜ 0 ⎟ ⎪ ⎝ h ⎠ ∂θ ω h ⎝ ∂ r ⎠ ⎪ 2 ν f ⎛ ∂2w ⎞ ⎛ R ⎞ ⎛ X0 ⎞ ∂ p ⎪ + − − iw ⎜ ⎟=0 ⎜ ⎟ ⎜ ⎟ ⎪ ωh2 ⎝ ∂ r 2 ⎠ ⎝ H ⎠ ⎝ h ⎠ ∂z ⎩
[7.342]
The boundary conditions [7.339] become: iX 0 iX cos θ ; v ( 0, θ , z ) = − 0 sin θ ; w ( 0, θ , z ) = 0 h h u (1, θ , z ) = v (1, θ , z ) = w (1, θ , z ) = 0 ; p (θ ,0 ) = p (θ ,1) = 0
u ( 0, θ , z ) =
[7.343]
Solutions are sought in the following form: u ( r , z ) cos θ
; v ( r , z ) sin θ
; w ( r , z ) cos θ
;
p ( z ) cos θ
By substituting [7.344] into equations [7.342], one obtains:
[7.344]
Energy dissipation by the fluid
⎧ ∂u ∂w +v + =0 ⎪ ∂ ∂z r ⎪ ⎪ ν f ⎛ ∂2v ⎞ ⎪ ⎛ X0 ⎞ − − iv p ⎨ ⎜ ⎟=0 ⎜ ⎟ ωh2 ⎝ ∂ r 2 ⎠ ⎝ h ⎠ ⎪ 2 ⎪ 2 ν ⎪iw + ⎛⎜ R ⎞⎟ ⎛⎜ X 0 ⎞⎟ ∂ p − f ⎜⎛ ∂ w ⎟⎞ = 0 2 2 ⎪⎩ ⎝ H ⎠ ⎝ h ⎠ ∂ z ωh ⎝ ∂ r ⎠
693
[7.345]
Whence the general solutions of the two last equations: v ( r , z ) = A− e kr + A+ e − kr −
iX 0 p h
w ( r , z ) = B− e kr + B+ e − kr +
iX 0 ⎛ R ⎞ dp ⎜ ⎟ h ⎝ H ⎠ dz
[7.346]
2
where the dimensionless and complex wave number is defined as: k = (1 + i )
ωh2 = (1 + i ) Sν / 2 2ν 0
[7.347]
The constants can be obtained by solving the boundary relations: −iX 0 iX = A− + A+ − 0 p R h
A− e k + A+ e − k −
;
iX 0 p=0 h
2
B− + B+ +
2
iX 0 ⎛ R ⎞ dp =0 ; ⎜ ⎟ h ⎝ H ⎠ dz
B− e k + B+ e − k +
iX 0 ⎛ R ⎞ dp =0 ⎜ ⎟ h ⎝ H ⎠ dz
[7.348]
After a few manipulations we arrive at: A− =
iX 0 p 1 − e − k h ek − e− k
B− =
−iX 0 ⎛ R ⎞ dp 1 − e − k ⎜ ⎟ h ⎝ H ⎠ dz e k − e − k
;
A+ =
iX 0 p e k − 1 h ek − e − k
2
2
; B+ =
−iX 0 ⎛ R ⎞ dp e k − 1 ⎜ ⎟ h ⎝ H ⎠ dz e k − e − k
[7.349]
The tangential and axial components of the velocity field are then expressed as: v (r , z ) =
(
)
⎧ sinh ( k (1 − r ) + sinh ( kr ) ⎫⎪ iX 0 ⎪ p ( z ) ⎨ −1 + ⎬ h sinh k ⎪⎩ ⎪⎭
(
)
sinh ( k (1 − r ) + sinh ( kr ) ⎫⎪ iX ⎛ R ⎞ dp ⎧⎪ w(r , z ) = − 0 ⎜ ⎟ ⎨ −1 + ⎬ h ⎝ H ⎠ dz ⎪ sinh k ⎪⎭ ⎩ 2
[7.350]
694
Fluid-structure interaction
Substituting [7.350] into the continuity equation given as the first relation of system [7.342], the radial gradient of the radial velocity component is found to be:
(
)
2 2 ∂ u −iX 0 ⎜⎛ sinh ( k (1 − r ) + sinh ( kr ) ⎟⎞ ⎛ ⎛ R ⎞ d p⎞ = −1 ⎜ p ( z ) − ⎜ ⎟ 2 ⎟ ∂r h ⎜ sinh k ⎟⎜ ⎝ H ⎠ d z ⎠⎟ ⎝ ⎠⎝
[7.351]
Integrating equation [7.351] with respect to r is immediate: u (r , z ) =
(
)
2 ⎞ 2 iX 0 ⎛⎜ cosh ( kr ) − cosh ( k (1 − r ) ⎛ ⎛ R ⎞ d p⎞ − kr ⎟ ⎜ p ( z ) − ⎜ ⎟ +C 2 ⎟ kh ⎜ sinh k ⎟ ⎜⎝ ⎝ H ⎠ d z ⎟⎠ ⎝ ⎠
[7.352]
Applying now the conditions at the walls, we arrive at the following relations: u (1, z ) = 0 ⇒ C =
iX 0 ⎛ 1 − cosh k ⎜k + kh ⎝ sinh k
2
2 ⎞⎛ ⎛ R ⎞ d p⎞ − p z ( ) ⎜ ⎟⎜ ⎜ ⎟ 2 ⎟ ⎝ H ⎠ d z ⎟⎠ ⎠⎝
2
2 iX k sinh k ⎛R⎞ d p = u ( 0, z ) = 0 ⇒ p ( z ) − ⎜ ⎟ 2 h 2 (1 − cosh k ) + k sinh k ⎝H ⎠ dz
[7.353]
The differential equation whih governs the pressure field is further transformed into: 2
2 ⎛R⎞ d p = p(z ) −⎜ ⎟ 2 ⎝H ⎠ dz
1
[7.354]
⎛k ⎞ 2 1 − tanh ⎜ ⎟ k ⎝2⎠
which is integrated as: 1
p ( z ) = p− eη z + p+ e −η z + 1−
[7.355]
⎛k ⎞ 2 tanh ⎜ ⎟ k ⎝2⎠
where η = H / R . Finally the constants in the solution [7.355] are determined by applying the axial boundary conditions p ( 0 ) = p (1) = 0 and the pressure field is finally obtained as: ⎛ ⎞⎛ ⎛ H (1 − z ) ⎞ ⎛ Hz ⎜ ⎟ ⎜ sinh ⎜ ⎟ + sinh ⎜ R 1 ⎝ R ⎝ ⎠ ⎟⎜ p(z ) = ⎜ ⎜ ⎛ k ⎞ ⎟⎜ ⎛H⎞ 2 sinh ⎜ ⎟ ⎜⎜ 1 − tanh ⎜ ⎟ ⎟⎟ ⎜⎜ ⎝R⎠ k ⎝ 2 ⎠ ⎠⎝ ⎝
⎞ ⎟ ⎠
⎞ ⎟ − 1⎟ ⎟ ⎟⎟ ⎠
[7.356]
The dimensioned generalized force exerted on the tube vibrating locally at the uniform amplitude X0 is:
Energy dissipation by the fluid
Q = −ω 2 ρ f π R 2 H
R h
1
⌠ X 0 ⎮⎮ ⌡0
695
⎛ 2R ⎛H ⎞ ⎞ tanh ⎜ ⎟ − 1⎟ ⎜ R H ⎝ 2R ⎠ ⎠ p ( z ) dz = −ω 2 ρ f π R 2 H ⎝ X0 h ⎛k ⎞ 2 1 − tanh ⎜ ⎟ k ⎝2⎠
[7.357] which is finally expressed in the compact form: Q = ω 2 M a Cν X 0
[7.358]
where the inviscid added mass coefficient is: M a = ρ f π R2 H
R ⎛ 2R ⎛ H ⎞⎞ 1− tanh ⎜ ⎟⎟ ⎜ h⎝ H ⎝ 2R ⎠ ⎠
[7.359]
and the viscous coefficient Cν is: ⎛k ⎞ 2 Cν = 1 − tanh ⎜ ⎟ k ⎝2⎠
−1
[7.360]
It can be verified that Cν is in full agreement with the formula [7.323] established in subsection 7.3.2.2, in the context of flexible plates. Starting from formula [7.360], the result [7.323] can be recovered by replacing the dimensionless quantities by the dimensioned ones and recalling that h is the total thickness of the fluid layer in the present problem whereas it stands for half of the thickness in the flexible plate problem. On the other hand, the added mass coefficient [7.359] is in full agreement with the coefficient already calculated in Chapter 3, formula [3.235]. It corresponds to H times the mass coefficient per unit tube length resulting from the strip model wheighted by the leakage effect associated to the pressure release conditions at z = 0 and z = H. 7.3.2.5 Application to fluid induced damping of multisupported tubes Figure7.67 sketches a portion of a heat exchanger tube loosely maintained by several support plates which present small gaps. The stainless steel tube contains a liquid, typically hot water, which heats an external liquid at smaller temperature by thermal conduction trough the tube wall. Both fluids are in steady motion. The upper sketch shows an idealized configuration where the tube is centred with respect to all the support plates resulting, in principle at least, in a constant effective gap between the supports and the tube. Such a configuration is highly unrealistic as the unsupported tube is extremely flexible and easily bent more than that is allowed by the gaps, even if the forces exerted on it are very small, as it is unavoidable in reality, due to the external flow in particular. The lower sketch is another idealized configuration whih turns to be more realistic, in which the tube is in contact with the
696
Fluid-structure interaction
Figure 7.67. Heat exchanger tube provided with multiple loose support plates: actual and idealized configurations
right-hand and left-hand supports and loose at the central support. Contact conditions are idealized as pinned supports to allow one to keep in the linear domain. Numerical data is as follows: Tube span between two plates L = 1 m, tube radius R = 1 cm, wall thickness e = 1 mm, plate thikness H = 3 cm, width of the annular space h = 0.2 mm. The equivalent mass per unit tube length including the added mass is me = 1.6 kg/m. The first bending mode corresponding to the pinned-pinned support conditions with a tube span 2L is f1 35 Hz . Properties of such a mode are significantly modified by fluid-structure coupling at the loose supports. First the added mass coefficient related to the annular gap is not negligible mag 0.45 kg if compared with the modal mass me / 2 = 0.8 kg . Of much more practical interest, is the fact that viscous damping is also found to be significant though its value is very sensitive to temperature as is the viscosity coefficient. Indeed, the kinematic viscosity of water at 20 °C is about 10−6 m 2s-2 ; it decreases to about 10−7 m 2s-2 at 300 °C. Accordingly the modal damping ratio is found to decrease from about 24 % to 3% when passing from the cold conditions to the normal working conditions of the plant. To conclude on the subject, the main interest of the calculation made here is to demonstrate that fluid-induced damping can be important as soon fluid is highly confined, even locally, and even if viscosity of the fluid is small, as is the case for hot water. Nevertheless, more refined analyses are clearly required to predict realistic damping values of heat exchanger tubes, where the support conditions can vary in large proportions from one support to the next and where various nonlinear effects are likely to occur due to vibration amplitudes of the order of the gap, in particular
Energy dissipation by the fluid
697
impacts and sliding against the supports. The reader interested in the subject can be reported to the specialized literature, in particular [PET 86a,b] and [ESM 92]. 7.4. Dissipation in acoustic waves Several mechanisms are responsible for sound absorption. All of them are related to irreversible transfers of momentum and heat that occur due to friction and thermal conductivity. Actually, when compressibility of the fluid is accounted for, the Stokes tensor [1.41] as introduced in Chapter 1 subsection 1.2.2.7, becomes insufficient to model all the relevant features of molecular friction, as further discussed a little later. Nevertheless, for the sake of simplicity it is preferable to proceed step by step in the presentation, starting from the simplest model which relies on the Stokes viscous stresses. 7.4.1
Viscous dissipation
7.4.1.1 Plane unconfined waves The sound waves are modelled here starting from the linearized Navier-Stokes equations [1.54] rewritten as: 1 =0 + div p ρ X f f c 2f [7.361] 1 1 ⎛ ⎞ grad p − ν f ⎜ ΔX f + grad divX f ⎟ = 0 Xf + 3 ρf ⎝ ⎠
(
)
Particular solutions are searched in the form of harmonic plane waves, travelling along the Ox direction. The particle velocity field X f and the gradient of it are thus one-dimensional and also directed along Ox. As the wave is harmonic, X f is conveniently described by its complex amplitude written as: iω X f ( x )eiω t = W ( x )eiω t i
[7.362]
where the pulsation ω is assumed here to be a real and positive quantity. By eliminating pressure between the two equations of [7.361], the following wave equation is obtained: 2 ⎛ 2 4 ⎞d W 2 ⎜ c f + iων f ⎟ 2 + ω W = 0 3 dx ⎝ ⎠
[7.363]
Solution is immediate, leading once more to the linear superposition of an outgoing and an ingoing wave written as: W ( x; ω t ) = A+ ei (ωt −kν x ) + A− ei (ωt + kν x )
[7.364]
The only difference with the conservative case is that due to viscosity, the wave number, denoted kν , becomes complex:
698
Fluid-structure interaction
ω
kν = cf
k
=
4 iων f 1+ 3 c 2f
[7.365]
2 4k νf 1+ i 3 ω
where k = ω / c f designates here the real wave number. As could be anticipated, the imaginary part is related to viscosity and by letting ν f vanish the conservative case is recovered. Moreover, viscosity introduces some dispersion into the wave propagation, which is a mere consequence of the fact that viscous friction force depends on frequency. It is also of interest to transform the result [7.365] in such a manner as to express the dimensionless dissipative coefficient as a ratio involving the two length scales wich are physically pertinent to the problem, namely the real wavelength λ = 2π / k which characterises the distance travelled per cycle by the wave and ην wich characterizes the space attenuation due to viscosity. Accordingly [7.365] is rewritten as: kν =
2π 2 ( 2π ) ⎛ ην ⎞ λ 1+ i ⎜ ⎟ 3 ⎝λ ⎠ 2
[7.366]
2
In most cases, ην is much less than λ which allows one to simplify the relations [7.365] and [7.366] as: ⎛ω kν ⎜ ⎜c ⎝ f
⎞⎛ 2ων f ⎟⎟ ⎜⎜ 1 − i 3c 2f ⎠⎝
2 ⎛ ⎞ (ην k ) ⎞ = k − iα ⎟ ⎟⎟ = k ⎜ 1 − i ν ⎜ 3 ⎟⎠ ⎠ ⎝
[7.367]
where αν is the attenuation coefficient due to viscosity, given by: αν =
2ω 2ν f
[7.368]
3c 3f
The simplified version of the waves [7.364] follows as: W ( x ; ω t ) = e −αν x
( A+ e (
i ω t − kx )
+ A− e (
i ωt + kx )
)
[7.369]
The outgoing and ingoing waves travel essentially at the same speed as in the conservative case and are progressively attenuated on a characteristic length distance Lν = 1/ αν which is far much larger than the wavelength. In air at STP the
attenuation length is considerable, Lν ( km ) 108 f −2 ; in water it is even much larger by three orders of magnitude. In reality, as shortly discussed later in subsection 7.3.4, other mechanisms are more efficient than viscosity to absorb sound in unconfined spaces.
Energy dissipation by the fluid
699
7.4.1.2 Importance of fluid confinement in viscous dissipation It can be easily understood that dissipation due to viscosity increases if fluid confinement is enhanced since visous forces are proportional to the gradient of fluid velocity. As an elementary example, let us consider a fluid layer of thickness L, delimited by two parallel and perfectly reflecting planes. By adjusting the constant in the general solution [7.369] to fulfil the boundary conditions, one obtains complex one-dimensional acoustic modes of complex natural pulsations: ωn( ) = ν
nπ c0 ⎛ 2i nπν f ⎜1 + 3 Lc f L ⎜⎝
⎞ ⎟⎟ ⎠
[7.370]
The modal damping ratio follows as ςn =
2 nπν f 3 L
[7.371]
As could have been anticipated, the damping ratio is proportional to the rank of the mode and inversely proportional to the gap between the walls. In other terms, it is proportional to the modal wave number. It is also pertinent to associate a modal Stokes number to the fluid mode, which is of the type: S(n ) = ν
nπ c f λn2 νf
=
π c f L2 nν f
[7.372]
By comparison to [7.370], in unbounded fluid the appropriate length scale for building the pertinent Stokes number is the wavelength of the travelling wave, whence: Sν =
2 2 ωλ 2 4π c f = νf ων f
[7.373]
Finally, if dissipation through viscosity is considered for plane waves in a tube, one is tempted to define the Stokes number by using the tube radius or diameter D as the pertinent scale length. Nevertheless, the relevant scaling factor for dissipation is in fact the thickness of the boundary layer which develops at the tube wall. As shown in the next subsection, depending on the tube diameter, it can be much smaller than or of the same order of magnitude as D. 7.4.1.3 Plane waves confined in a circular cylindrical tube Having addressed the incompressible case in subsection 7.3.2.1, we now turn to the case of plane sound waves within a tube filled with a viscous fluid, by which one means that the gradient of the pressure field has only a longitudinal or axial component. In contrast with the inviscid case, the associated fluid velocity field can not however be strictly plane because it must vanish at the wall to accommodate the no-slip condition and differs from zero elsewhere since fluid is assumed to oscillate. To analyse the problem further, it is convenient to consider a circular cylindrical
700
Fluid-structure interaction
tube of length L and radius R, terminated at both ends by a conservative impedance, a pressure node for instance. The fluid velocity field is also of revolution and is approximated as the axial form: X f = W ( x, r )i [7.374] Pertinence of this choice can be supported as follows. First, it is expected that the viscous counterpart of plane sound waves is also predominantly described by an axial velocity field, driven by the axial gradient of pressure. Then, any radial component U of the fluid velocity can develop as a consequence of viscous shear solely, which is mainly tangential and usually confined at the vicinity of the wall in a thin boundary layer ην . The system of equations [7.361] particularises into: iω p ( x ) = − ρ f ce2
∂W ∂x
[7.375]
⎛ 4 ∂ 2W ∂ 2W 1 ∂W ⎞ 1 ∂p −ν f ⎜ + 2 + iωW + ⎟=0 2 ∂r ρf ∂x r ∂r ⎠ ⎝ 3 ∂x
where again ce is the equivalent speed of sound in the tube taking into account elasticity of the walls. By eliminating pressure between these two equations, the following wave equation is obtained: ⎛ 4iων f ⎞ ∂ 2W ⎛ ∂ 2W 1 ∂W ω 2W + ce2 ⎜ 1 + ⎟ 2 + iων f ⎜ 2 + 2 3ce ⎠ ∂ x r ∂r ⎝ ∂r ⎝
⎞ ⎟=0 ⎠
[7.376]
We seek for solutions in separated form W = w( x )u ( r ) , which leads to: ⎛ 4iων f ⎞ w′′ ⎛ u′′ u′ ⎞ ω 2 + ce2 ⎜ 1 + ⎟ + iων f ⎜ + ⎟ = 0 2 3ce ⎠ w ⎝ u ru ⎠ ⎝
[7.377]
The axial equation is the same as [7.361] solutions are thus again of the form [7.362]. Finally, the boundary conditions at tube ends imply that: dw ⎛ nπ x ⎞ = 0 ⇒ wn ( x ) = cos ⎜ ⎟ n = 0,1, 2... dx 0, L ⎝ L ⎠
[7.378]
According to the results [7.378], the mode shapes of the axial fluid velocity are thus found to be the same as in the inviscid case. The n = 0 mode stands for the axial motion of the fluid column which behaves as an incompressible fluid. Hence fluid velocity is uniform in the axial direction and pressure is nil. Substituting the axial modes shapes [7.238] into [7.377], the radial variation of the axial velocity is found to verify the following Bessel equation of zero order: ⎧⎪ 2 ⎛ nπ c f ⎞2 ⎛ 4iων f ⎨ω − ⎜ ⎟ ⎜1 + 3ce2 ⎝ L ⎠ ⎝ ⎩⎪
⎞ ⎫⎪ ⎟ ⎬ u + iων f ⎠ ⎭⎪
⎛ ′′ u ′ ⎞ ⎜u + ⎟ = 0 r⎠ ⎝
[7.379]
Energy dissipation by the fluid
701
The physically meaningful solution follows as: u ( r ) = AJ 0 ( kr ) ; k 2 =
1 iων f
⎧⎪ 2 ⎛ nπ c f ⎞ 2 ⎛ 4iων f ⎨ω − ⎜ ⎟ ⎜1 + 3ce2 ⎝ L ⎠ ⎝ ⎩⎪
⎞ ⎫⎪ ⎟⎬ ⎠ ⎭⎪
[7.380]
The no-slip wall condition quantifies the acceptable wave numbers as: J 0 ( kR ) = 0 ⇒ β1 = k1 R = 2.4083..., β 2 = k2 R = 5.5208..., etc.
[7.381]
Whence, the equation giving the modal pulsations: ⎛ 4 ⎛ nπ ⎞ 2 β m2 ⎞ ⎛ nπ ce ⎞ 2 ω 2 + iων f ⎜ ⎜ − − =0 ⎜ 3 ⎝ L ⎟⎠ R 2 ⎟⎟ ⎜⎝ L ⎟⎠ ⎝ ⎠
[7.382]
However, it is natural to restrict the mathematical solutions to the special case m = 1, otherwise W ( x, r; ω t ) would change of sign depending on the radial distance from the tube axis, which is clearly not appropriate for a plane wave approximation. Therefore the modal damping ratios are found to be: ςn =
ν f ⎛ L ⎞ ⎛ ⎛ nπ ⎞2 β1 ⎞ ⎜ ⎟ ⎜⎜ 4 ⎜ ⎟ + 3 ⎟⎟ 6 ⎝ nπ ce ⎠ ⎝ ⎝ L ⎠ R⎠
2
n = 1, 2,...
[7. 383]
The result [7.383] clearly indicates that dissipation is due to viscous friction at the wall since it is est proportiona to the tube length and its dominating term is proportional to the reciprocal of the cross-sectional area. Notice however that these damping values are usually small except in the case of capillary tubes. Let us consider a tube filled with water: L = 10 m, R = 1 cm, ce = 1 400 m/s, ν f = 10−6 m 2s-1 , one obtains ς 1 = 3.910−4 ; in air at STP one would obtain ς 1 = 10−3 . 7.4.2
Miscellaneous dissipative mechanisms in acoustic waves
Viscosity is only one of a number of molecular processes which lead to sound absorption. There is no place to review here the miscellaneous processes whose relative importance may greatly vary depending on the particular physical context of the problem. As such mechanisms are well documented in many excellent textbooks in Acoustics, suffice it to mention them briefly, reporting the reader to a few pertinent references. However, it is of special interest to say a little more on the thermodynamical aspect of the problem in gases since the subject has been already mentioned in Chapter 1 in the context of the viscous Stokes tensor and in Chapter 4, in the context of adiabatic and isothermal sound propagation. It is found appropriate to start by describing the heat conduction mechanism which results from the irreversible conduction of thermal energy driven by the temperature gradient associated with the wave condensations and rarefactions, as already explained in Chapter 4, subsection 4.4.2.
702
Fluid-structure interaction
7.4.2.1 Heat conduction and thermoacoustic coupled waves Coupling between heat conduction and sound waves can be decribed by complementing the linearized Navier-Stokes equations by an equation which describes compressibility of the gas and another to account for the rate of change in entropy. Following the presentation given by Morse and Ingard [MOR 86], except that viscous terms are discarded here since they were already described earlier, relations [1.54] are rewritten in the context of free waves as: ∂ρ + ρ 0 div X f = 0 ∂t [7. 384] ρ 0 X f + grad p = 0 Compressibility law in [1.54] must be modified to account for temperature changes. The thermomecanical law is written as: ⎛ ∂ρ ⎞ ⎛ ∂ρ ⎞ γ 1 ρ = ⎜ ⎟ p+⎜ ⎟ θ = 2 ( p − αθ ) = 2 ( p − αθ ) ∂ ∂ P P c c ⎝ ⎠T ⎝ ⎠P i a
[7. 385]
where ci = ca γ is the isothermal and ca the adiabatic speed of sound in a perfect gas, in agreement with relation [1.33] and α denotes the rate of increase of pressure with temperature at constant volume as defined by formula [A1-13] in Appendix A1. The coefficient of the small temperature change θ is obtained by using formulas [A1-11] and [A1-15]. In passing it may be shown with the aid of formulas [A1-13] and [A1-31] that an isovolumetric process corresponds to p − αθ = 0 . On the other hand, with the aid of formulas [A1-21], [A1-27] the change in entropy for a perfect gas can be written as: ⎛ θ ⎛ γ −1⎞ p ⎞ ⎛ ∂S ⎞ ⎛ ∂S ⎞ s=⎜ ⎟ p+⎜ ⎟ θ = CP ⎜ − ⎜ ⎟ ⎟ P P ∂ ∂ ⎝ ⎠T0 ⎝ ⎠ P0 ⎝ T0 ⎝ γ ⎠ P0 ⎠
[7. 386]
In agreement with formula [A1-42], adiabatic sound propagation is found to be isentropic. The rate of change in entropy is approximated by the reversible law [A1.64], rewritten here as: ρ0
ds κ H = Δθ dt T0
[7. 387]
where ρ 0 denotes the density and T0 the temperature of the fluid at the state of reference, while s denotes here the small change in entropy related to the small change in temperature θ. Such a simplified law adopted in particular in [MOR 86] assumes that heat transfer can be treated as a dissipative perturbation of a nearly adiabatic process. Eliminating the fluid velocity between the mass and momentum equations [7.384] and substituting [7.385] for the fluctuating density, we obtain a first thermoacoustic equation, which turns out to stand for the isothermal wave equation complemented by a thermal term:
Energy dissipation by the fluid
Δp −
1 ∂ 2 p α ∂ 2θ + =0 ci2 ∂ t 2 ci2 ∂ t 2
703
[7. 388]
A second thermoacoustic equation is obtained by substituting the fluctuating entropy [7.386] into the equation of evolution [7.387], which gives an equation of first order in time which turns out to stand for the heat diffusion equation complemented by a pressure term: ⎛ γ − 1 ⎞ ∂p ∂θ − χ H Δθ − ⎜ =0 ⎟ ∂t ⎝ γ R ρ0 ⎠ ∂ t
[7. 389]
where the thermal diffusivity χH was already defined, see relation [4.227]. Harmonic thermoacoustic plane waves are of the type: ⎡ p0 ⎤ i (ωt − kx ) ⎢θ ⎥ e ⎣ 0⎦
[7. 390]
Substituting [7.390] into the system of equations [7.388] and [7.389], we arrived at the coupled wave matrix equation: ⎡ ki2 − k 2 −α ki2 ⎤ ⎢ ⎥ ⎡ p0 ⎤ ⎡0⎤ ⎢ −ik ⎛ γ − 1 ⎞ ik + χH k 2 ⎥ ⎢θ ⎥ = ⎢ 0⎥ i ⎢ i ⎜⎝ αγ ⎟⎠ ⎥⎣ 0⎦ ⎣ ⎦ ci ⎣ ⎦
[7. 391]
where ki is the dimensionless isothermal wave number ω / ci and relation [A1-31] is used. The complex wave numbers of the coupled waves are given by solving the dispersion equation: Λk 4 + iki (1 + iΛki ) k 2 − i
ki3 =0 γ
[7. 392]
where Λ is a characteristic length for thermal diffusion defined as: Λ=
χH ci
[7. 393]
Λ is of the order of the mean free path of the molecules, as already shown in Chapter 4 (see formula [4.237]). Thus it is of interest to discuss the approximated solutions of equation [7.392] which hold in the long wavelength range, such that Λ is much shorter than the acoustic wavelengths of interest. It is also instructive to proceed by increasing progressively the degree of refinement of the approximations made. If equation [7.392] is drastically simplified, neglecting all terms containing the short distance Λ, the adiabatic acoustic solution is recovered. More interesting is to proceed by expanding the discriminant of the biquadratic equation [7.392] to the first order with respect to the small dimensionless parameter Λki . Calculation presents no difficulty, except that much care is needed to perform it. Thus the main
704
Fluid-structure interaction
steps are outlined here, for the sake of clarity. The discriminant of [7.392] is first written as: ⎛ ⎛ 2⎞ 2⎞ Δ = − ki2 ⎜ 1 + 2i Λki ⎜ 1 − ⎟ − ( Λki ) ⎟ ⎝ γ⎠ ⎝ ⎠
Expanding
[7. 394]
Δ to the first order in Λki gives:
⎛ ⎛ 2 ⎞⎞ Δ iki ⎜ 1 + i Λki ⎜ 1 − ⎟ ⎟ ⎝ γ ⎠⎠ ⎝
If this first order approximation of following results arise:
[7. 395] Δ is used to solve equation [7.392], the
2
⎛ω⎞ ω k+2 = ka2 = ⎜ ⎟ ⇒ k+ = c c a ⎝ a⎠ ⎛ ω ⎞ ω ⎛k ⎞ k − i ⎜ i ⎟ = −i ⎜ ⎟ ⇒ k− = (1 − i ) Λ χ χH 2 ⎝ ⎠ ⎝ H ⎠
[7. 396]
2 −
Therefore, it turns out that to the first order in Λki , the two waves described by the coupled thermoacoustic equations are an undamped adiabatic sound wave, so unchanged with respet to the conservative model, and a evanescent thermal wave, characteriszed by the skin depth: ηH =
2χH ω
[7. 397]
which can be interpreted as the depth of a thermal boundary layer for heat conduction. Furthermore, it is of interest to build the ratio of the viscous to the termal skin depth, which is a measure of the relative importance of viscous to heat conduction dissipation. With the aid of the expression [7.289], this dimensionless quantity is found to be: ην = ηH
νf χH
= Pr
[7. 398]
Pr is known as the Prandtl number, which is met here in the context of oscillatory motions. Finally, to bring out the thermal loss in the acoustic wave due to thermal conduction, it is found necessary to expand Δ to the second order in Λki , which gives: 2 ⎛ ⎛ 2⎞ ⎛ Λk ⎞ ⎞ Δ iki ⎜ 1 + i Λki ⎜ 1 − ⎟ + 2 (1 − γ ) ⎜ i ⎟ ⎟ ⎜ ⎝ γ⎠ ⎝ γ ⎠ ⎟⎠ ⎝
[7. 399]
Energy dissipation by the fluid
705
With the aid of [7.399], the following complex wave number for the sound wave is found: ⎛ (γ − 1) χH ω 2 χ ω⎞ ω k +2 = ka2 ⎜ 1 + i (1 − γ ) H2 ⎟ ⇒ k+ = − i 2ca3 ca ⎠ ca ⎝
[7. 400]
In the same manner as for dissipation due to viscosity, dissipation due to thermal conduction is characterized by an attenuation length, defined here as: LH =
2ca3 (γ − 1) χH ω 2
[7. 401]
Viscous and thermal dissipation are found to be very small in ordinary fluids like air, then the two processes can be conveniently accounted for in a single formula which adds both effects. It is expressed in terms of the following absorption coefficient, broadly known as the classical absorption coefficient, defined as: α classical =
ω 2ν f ⎛ 4 γ − 1 ⎞ 1 1 + = ⎜ + ⎟ Lν LH 2ca3 ⎝ 3 Pr ⎠
[7. 402]
To fix the ideas, for air at STP, α classical 1.410−11 f 2 , where f denotes the frequency in Hz. To conclude this subsection, it is also of interest to mention that temperature can be eliminated between equations [7.388] and [7.389] to produce a single wave equation in terms of pressure solely, which turns out to be remarkable for its formal simplicity and physical content. Equation [7.389] is first transformed into: α ∂ 2θ αχH ∂Δθ ⎛ γ − 1 ⎞ ∂ 2 p = 2 +⎜ ⎟ ci2 ∂ t 2 ci ∂t ⎝ ca2 ⎠ ∂ t 2
[7. 403]
Then, substituting the intermediate result [7.393] into equation [7.388] gives: 1 ∂2 p αχ ∂Δθ − Δp − 2H =0 2 2 ca ∂ t ci ∂t
[7. 404]
Deriving equation [7.404] with respect to time and taking the Laplacian of equation [7.388] gives the following wave equation: ⎛ ∂2 p 2 ⎞ ∂ ⎛ ∂2 p 2 ⎞ ⎜ 2 − ca Δp ⎟ − χH Δ ⎜ 2 − ci Δp ⎟ = 0 ∂t ⎝ ∂t ⎠ ⎝ ∂t ⎠
[7.405]
which is remarkable for its formal simplicity. The adiabatic case is recovered by letting χH = 0, which means that there is no diffusion of heat. At the other extreme, the isothermal case is recovered by letting χH tends to infinity, which means that heat is immediately removed from the wave condensations and rarefactions.
706
Fluid-structure interaction
7.4.2.2 Relaxation mechanisms When comparative data on calculated and observed values of the classical absorption coefficient are listed for various gases and liquids, a satisfactory agreement is found for many fluids, but large discrepancies are also found to occur in many others. Namely, classical absorption fits reality in the case of monoatomic gases, highly viscous fluids like glycerine and highly conducting metals. However, it underestimates dissipation in polyatomic gases and in most common liquids. The subject is detailed in many excellent textbooks in Acoustics, and the reader will find enlightening information in particular in [MOR 86], [LAN 59] concerning the second coefficient of viscosity, or bulk viscosity coefficient, in polyatomic gases, and in [TEM 01], [KIN 00], [BLA 00] concerning other relaxation mechanisms, which are of special importance in many fluids such as sea water, atmospheric air etc. As a definition, in a relaxing fluid, the change in pressure depends not only on the local and instantaneous value of density and temperature - as was assumed to be the case throughout the present book up to here - but also on the rate of change of such quantities. As a first example, let us consider the second coefficient of viscosity introduced together with the Stokes stress tensor in subsection 1.2.2.7, where it was explained that a shear viscous stress σ zx is a resistive force which tends to smooth out the difference in tangential velocity of two adjacent fluid layers by accelerating the slowest layer and decelerating the fastest. At the molecular level, the process is described as a consequence of the fact that particles are continually jumping from one layer to the other, while transferring momentum and energy through collisions, as already outlined in subsection 4.4.3. Some amount of time is inevitably required for such a molecular adjustement, which can be referred to viscous relaxation. In monoatomic gases, only translational degrees of freedom are involved in the process. But, in polyatomic gases, other degrees of freedom are also to be considered, related to the various rotation and vibrations modes of the molecule. When a fluid element is thrown out of equilibrium, by a small change in pressure for instance, the amounts of energy laying in the various possible modes of motion differ from the equilibrium values and the discrepancies are progressively diminished through molecular collisions. The number of collisions and the time lag necessary to reach the new equilibrium values depend on the specific mode considered. A net consequence of such molecular processes is that the thermodynamic pressure differs from the mechanical pressure, in contrast with the Stokes assumption. In terms of viscosity coefficients, this means that the second viscosity coefficient is no more related to the first one through equation [1.40]. On the other hand, it can be demonstrated that sound absorption is very significantly increased as frequency nearly coincides with a characteristic relaxation frequency of the fluid. The latter quantity is defined as the reciprocal of the characteristic time necessary for a “vibration mode” to recover equilibrium. On the specific subject the interested reader can be reported to [HER 66] and [BHAT 67].
Energy dissipation by the fluid
707
Other phenomena presenting similarities with the molecular processes described just above are chemical relaxation processes. Finally, when a fluid contains inhomogeneities such as suspended bubbles or particles etc. additional sound damping is observed, due to the fact that the equilibrium between each inhomogeneity and the surrounding fluid is perturbed by the sound wave and some relaxation time is required for adjustement to the new equilibrium conditions.
Appendix A1
A few elements of thermodynamics A1. Thermodynamic refresher The aim of this appendix is to provide the reader with a short presentation of the few concepts and relationships of thermodynamics which are very necessary for understanding a few important aspects of the behaviour of compressible fluids, in acoustics in particular. The presentation given here is inspired for one part from that found in [TEM 01] and for the other from that found in [BLA 00]. It is recalled first that, as an empirical fact, to describe the thermodynamical state of a given quantity, a unit mass for instance, of a fluid containing a single chemical species, two independent fields are necessary and only two. Which two is immaterial, just like the nature of the generalized coordinates used to describe the dynamical state of a mechanical system is indifferent. Thermodynamics is grounded on three basic laws, the first two are however sufficient for our limited needs. A1.1
Law of energy conservation
A unit mass of a fluid is assumed to change from one equilibrium state to another, because a certain amount of heat Q is added to it and because it performs some amount of work W. Then the first law of Thermodynamics states that the internal energy of the fluid is changed by an amount Δ E given by: ΔE = Q + W
[A1.1]
Of course E, Q and W stand for energies per unit mass, as all the energetic quantities which follow. As an important point worth to be emphasized, E is a property of state so that Δ E depends only on the initial and the final state and not on the path followed from the initial to the final state, in contrast with Q and W. On the other hand, transition from one state to another can be irreversible or reversible. In a reversible transition, the system is assumed to pass continuously through a sequence of equilibrium states with no dissipation. To be reversible a compression or expansion must be performed sufficiently slowly as to preserve thermodynamic equilibrium at each intermediate state. Elemental work associated to the elemental compression dp of the unit mass of fluid is:
Appendices
dW = − Pdυ
709
[A1.2]
where υ = ρ −1 designates the specific volume of the fluid. Substituting [A1.2] into [A1.1], the amount of heat is suitably expressed as: dQ = dE + Pdυ
[A1.3]
On the other hand, it is also useful to define the enthalpy H as: H = E + Pυ
[A1.4]
So that the amount of heat can be also expressed as: dQ = dH − υ dP
[A1.5]
The change is called an adiabatic process if no heat flows into or out of the system (dQ = 0). For example, compression of a gas in a perfectly insulated tube is an adiabatic process. In this case, the change of internal energy of the gas is entirely due to the work done on it, or by it. As a consequence, temperature is changed since E depends on temperature. The specific heat for a path at constant volume is then defined as: ⎛ ∂E ⎞ Cυ = ⎜ ⎟ ⎝ ∂ T ⎠υ
[A1.6]
and for an isobaric path as: ⎛ ∂H ⎞ CP = ⎜ ⎟ ⎝ ∂ T ⎠P
[A1.7]
where T is the absolute temperature in °K. Thus, to produce an increase of dT in temperature for a fluid maintained at constant volume it is necessary to add an amount of heat dQ = Cυ dT per unit mass of fluid. If the process is made at constant pressure it is necessary to add dQ ′ = CP dT . It can be shown that dQ ′ is always larger than dQ , which means that the following ratio of the specific heats is larger than 1: γ =
CP >1 Cυ
[A1.8]
Another useful formula relating the specific heats is: ⎛ ∂υ ⎞ CP − Cυ = P ⎜ ⎟ ⎝ ∂ T ⎠P
[A1.9]
It can be proved by starting from the relations defining the specific heats. One obtains:
710
Fluid-structure interaction
⎛ ∂H ⎞ ⎛ ∂E ⎞ ⎛ ∂υ ⎞ CP = ⎜ ⎟ =⎜ ⎟ + P⎜ ⎟ ⎝ ∂ T ⎠P ⎝ ∂ T ⎠P ⎝ ∂ T ⎠P
[A1.10]
Hence, ⎛ ∂E ⎞ ⎛ ∂E ⎞ ⎛ ∂υ ⎞ CP − Cυ = ⎜ ⎟ −⎜ ⎟ + P⎜ ⎟ ∂ T ∂ T ⎝ ⎠P ⎝ ⎠υ ⎝ ∂ T ⎠P
Since E is a state variable, the derivatives with respect to temperature are path independent and [A1.9] is verified. A1.2
Compressibility and thermal expansion coefficients The coefficient of isothermal compressibility is defined as:
1 ⎛ ∂υ ⎞ 1 ⎛ ∂ρ ⎞ 1 κT = − ⎜ ⎟ = ⎜ ⎟ = 2 υ ⎝ ∂ P ⎠T ρ ⎝ ∂ P ⎠T ci
[A1.11]
Here we use notation κ T instead of κ (fT0 ) as in Chapter 1, formula [1–24], to alleviate notation. The coefficient of thermal expansion at constant pressure is: β=
1 ⎛ ∂υ ⎞ ⎜ ⎟ υ ⎝ ∂T ⎠ P
[A1.12]
The rate of increase of pressure with temperature at constant pressure is: ⎛ ∂P ⎞ α =⎜ ⎟ ⎝ ∂ T ⎠υ
[A1.13]
Considering a transformation at constant volume, we get: ⎛ ∂υ ⎞ ⎛ ∂υ ⎞ dυ ( T , P ) = ⎜ ⎟ dT + ⎜ ⎟ dP = 0 ∂ T ⎝ ⎠P ⎝ ∂ P ⎠T
[A1.14]
Substituting formulas [A1.11] and [A1.12], into [A1.14] the following relation between the three coefficients defined just above is obtained: α=
β κT
A1.3
[A1.15] Second law: entropy
The second law introduces another state property, termed entropy, by stating that if an elemental amount of heat dQ is added reversibly to a system, then entropy is varied by: dS =
dQ T
[A1.16]
Appendices
711
where S is the entropy per unit mass. A point of paramount importance is that as soon as the change of state is not totally reversible, entropy changes more than the amount which would be required to perform the same change by following a reversible process: dS >
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
Fluid-structure interaction
∫ 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
Fluid-structure interaction
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
Fluid-structure interaction
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
Fluid-structure interaction
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
Fluid-structure interaction
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
Fluid-structure interaction
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
Fluid-structure interaction
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
Fluid-structure interaction
⎧ 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
Fluid-structure interaction
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
Fluid-structure interaction
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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Fluid-structure interaction
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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
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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
