Fundamentals of Fluid Mechanics

and Transport Phenomena

Jean-Laurent Peube

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Fundamentals of Fluid Mechanics and Transport Phenomena

This page intentionally left blank

Fundamentals of Fluid Mechanics

and Transport Phenomena

Jean-Laurent Peube

First published in France in 2006 by Hermes Science/Lavoisier entitled Physique des coulements et des transferts in volumes 1 and 2 LAVOISIER, 2006 First published in Great Britain and the United States in 2009 by ISTE Ltd and John Wiley & Sons, Inc. Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address: ISTE Ltd John Wiley & Sons, Inc. 27-37 St Georges Road 111 River Street London SW19 4EU Hoboken, NJ 07030 UK USA

www.iste.co.uk www.wiley.com ISTE Ltd, 2009 The rights of Jean-Laurent Peube to be identified as the author of this work have been asserted by him in accordance with the Copyright, Designs and Patents Act 1988.

Library of Congress Cataloging-in-Publication Data Peube, J. L. [Physique des coulements et des transferts. English] Fundamentals of fluid mechanics and transport phenomena / Jean-Laurent Peube. p. cm. Includes bibliographical references and index. ISBN 978-1-84821-065-3 1. Fluid mechanics. 2. Transport theory. I. Title. TA357.P49813 2008 620.1'06--dc22

2008036553 British Library Cataloguing-in-Publication Data A CIP record for this book is available from the British Library ISBN: 978-1-84821-065-3

Printed and bound in Great Britain by CPI Antony Rowe Ltd, Chippenham, Wiltshire.

http://www.wiley.com

Table of Contents

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi

Chapter 1. Thermodynamics of Discrete Systems . . . . . . . . . . . . . . . . 1

1.1. The representational bases of a material system . . . . . . . . . . . . . . 1 1.1.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1.2. Systems analysis and thermodynamics . . . . . . . . . . . . . . . . . 8 1.1.3. The notion of state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.1.4. Processes and systems. . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.2. Axioms of thermostatics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.2.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.2.2. Extensive quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 1.2.3. Energy, work and heat . . . . . . . . . . . . . . . . . . . . . . . . . . 20

1.3. Consequences of the axioms of thermostatics . . . . . . . . . . . . . . . 21 1.3.1. Intensive variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 1.3.2. Thermodynamic potentials . . . . . . . . . . . . . . . . . . . . . . . . 23

1.4. Out-of-equilibrium states. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 1.4.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 1.4.2. Discontinuous systems . . . . . . . . . . . . . . . . . . . . . . . . . . 30 1.4.3. Application to heat engines . . . . . . . . . . . . . . . . . . . . . . . 45

Chapter 2. Thermodynamics of Continuous Media . . . . . . . . . . . . . . . 47

2.1. Thermostatics of continuous media. . . . . . . . . . . . . . . . . . . . . . 47 2.1.1. Reduced extensive quantities . . . . . . . . . . . . . . . . . . . . . . 47 2.1.2. Local thermodynamic equilibrium . . . . . . . . . . . . . . . . . . . 48 2.1.3. Flux of extensive quantities . . . . . . . . . . . . . . . . . . . . . . . 50 2.1.4. Balance equations in continuous media . . . . . . . . . . . . . . . . 54 2.1.5. Phenomenological laws . . . . . . . . . . . . . . . . . . . . . . . . . . 57

2.2. Fluid statics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 2.2.1. General equations of fluid statics . . . . . . . . . . . . . . . . . . . . 63

vi Fundamentals of Fluid Mechanics and Transport Phenomena

2.2.2. Pressure forces on solid boundaries. . . . . . . . . . . . . . . . . . . 68 2.3. Heat conduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

2.3.1. The heat equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 2.3.2. Thermal boundary conditions . . . . . . . . . . . . . . . . . . . . . . 72

2.4. Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 2.4.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 2.4.2. Molar and mass fluxes . . . . . . . . . . . . . . . . . . . . . . . . . . 77 2.4.3. Choice of reference frame . . . . . . . . . . . . . . . . . . . . . . . . 80 2.4.4. Binary isothermal mixture . . . . . . . . . . . . . . . . . . . . . . . . 85 2.4.5. Coupled phenomena with diffusion. . . . . . . . . . . . . . . . . . . 97 2.4.6. Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

Chapter 3. Physics of Energetic Systems in Flow. . . . . . . . . . . . . . . . . 101

3.1. Dynamics of a material point . . . . . . . . . . . . . . . . . . . . . . . . . 101 3.1.1. Galilean reference frames in traditional mechanics . . . . . . . . . 101 3.1.2. Isolated mechanical system and momentum. . . . . . . . . . . . . . 102 3.1.3. Momentum and velocity . . . . . . . . . . . . . . . . . . . . . . . . . 103 3.1.4. Definition of force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 3.1.5. The fundamental law of dynamics (closed systems) . . . . . . . . . 106 3.1.6. Kinetic energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

3.2. Mechanical material system . . . . . . . . . . . . . . . . . . . . . . . . . . 107 3.2.1. Dynamic properties of a material system . . . . . . . . . . . . . . . 107 3.2.2. Kinetic energy of a material system . . . . . . . . . . . . . . . . . . 109 3.2.3. Mechanical system in thermodynamic equilibrium: the rigid solid. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 3.2.4. The open mechanical system. . . . . . . . . . . . . . . . . . . . . . . 112 3.2.5. Thermodynamics of a system in motion . . . . . . . . . . . . . . . . 116

3.3. Kinematics of continuous media . . . . . . . . . . . . . . . . . . . . . . . 119 3.3.1. Lagrangian and Eulerian variables . . . . . . . . . . . . . . . . . . . 119 3.3.2. Trajectories, streamlines, streaklines . . . . . . . . . . . . . . . . . . 121 3.3.3. Material (or Lagrangian) derivative. . . . . . . . . . . . . . . . . . . 122 3.3.4. Deformation rate tensors . . . . . . . . . . . . . . . . . . . . . . . . . 129

3.4. Phenomenological laws of viscosity . . . . . . . . . . . . . . . . . . . . . 132 3.4.1. Definition of a fluid . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 3.4.2. Viscometric flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 3.4.3. The Newtonian fluid. . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

Chapter 4. Fluid Dynamics Equations . . . . . . . . . . . . . . . . . . . . . . . 151

4.1. Local balance equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 4.1.1. Balance of an extensive quantity G . . . . . . . . . . . . . . . . . . . 151 4.1.2. Interpretation of an equation in terms of the balance equation . . . 153

4.2. Mass balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 4.2.1. Conservation of mass and its consequences . . . . . . . . . . . . . . 154 4.2.2. Volume conservation . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

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4.3. Balance of mechanical and thermodynamic quantities . . . . . . . . . . 160 4.3.1. Momentum balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 4.3.2. Kinetic energy theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 164 4.3.3. The vorticity equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 4.3.4. The energy equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 4.3.5. Balance of chemical species . . . . . . . . . . . . . . . . . . . . . . . 177

4.4. Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 4.4.1. General considerations . . . . . . . . . . . . . . . . . . . . . . . . . . 178 4.4.2. Geometric boundary conditions . . . . . . . . . . . . . . . . . . . . . 179 4.4.3. Initial conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

4.5. Global form of the balance equations . . . . . . . . . . . . . . . . . . . . 182 4.5.1. The interest of the global form of a balance . . . . . . . . . . . . . . 182 4.5.2. Equation of mass conservation . . . . . . . . . . . . . . . . . . . . . 184 4.5.3. Volume balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 4.5.4. The momentum flux theorem . . . . . . . . . . . . . . . . . . . . . . 184 4.5.5. Kinetic energy theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 186 4.5.6. The energy equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 4.5.7. The balance equation for chemical species . . . . . . . . . . . . . . 188

4.6. Similarity and non-dimensional parameters . . . . . . . . . . . . . . . . . 189 4.6.1. Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

Chapter 5. Transport and Propagation. . . . . . . . . . . . . . . . . . . . . . . 199

5.1. General considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 5.1.1. Differential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 5.1.2. The Cauchy problem for differential equations . . . . . . . . . . . . 202

5.2. First order quasi-linear partial differential equations . . . . . . . . . . . 203 5.2.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 5.2.2. Geometric interpretation of the solutions . . . . . . . . . . . . . . . 204 5.2.3. Comments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206 5.2.4. The Cauchy problem for partial differential equations . . . . . . . 206

5.3. Systems of first order partial differential equations . . . . . . . . . . . . 207 5.3.1. The Cauchy problem for n unknowns and two variables . . . . . . 207 5.3.2. Applications in fluid mechanics . . . . . . . . . . . . . . . . . . . . . 210 5.3.3. Cauchy problem with n unknowns and p variables. . . . . . . . . . 216 5.3.4. Partial differential equations of order n . . . . . . . . . . . . . . . . 218 5.3.5. Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220 5.3.6. Physical interpretation of propagation . . . . . . . . . . . . . . . . . 223

5.4. Second order partial differential equations . . . . . . . . . . . . . . . . . 225 5.4.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 5.4.2. Characteristic curves of hyperbolic equations. . . . . . . . . . . . . 226 5.4.3. Reduced form of the second order quasi-linear partial differential equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 5.4.4. Second order partial differential equations in a finite domain . . . 232

viii Fundamentals of Fluid Mechanics and Transport Phenomena

5.4.5. Second order partial differential equations and their boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233

5.5. Discontinuities: shock waves . . . . . . . . . . . . . . . . . . . . . . . . . 239 5.5.1. General considerations . . . . . . . . . . . . . . . . . . . . . . . . . . 239 5.5.2. Unsteady 1D flow of an inviscid compressible fluid. . . . . . . . . 239 5.5.3. Plane steady supersonic flow . . . . . . . . . . . . . . . . . . . . . . 244 5.5.4. Flow in a nozzle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244 5.5.5. Separated shock wave . . . . . . . . . . . . . . . . . . . . . . . . . . . 248 5.5.6. Other discontinuity categories . . . . . . . . . . . . . . . . . . . . . . 248 5.5.7. Balance equations across a discontinuity . . . . . . . . . . . . . . . 249

5.6. Some comments on methods of numerical solution . . . . . . . . . . . . 250 5.6.1. Characteristic curves and numerical discretization schemes . . . . 250 5.6.2. A complex example . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 5.6.3. Boundary conditions of flow problems. . . . . . . . . . . . . . . . . 255

Chapter 6. General Properties of Flows . . . . . . . . . . . . . . . . . . . . . . 257

6.1. Dynamics of vorticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257 6.1.1. Kinematic properties of the rotation vector . . . . . . . . . . . . . . 257 6.1.2. Equation and properties of the rotation vector . . . . . . . . . . . . 261

6.2. Potential flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269 6.2.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269 6.2.2. Bernoullis second theorem . . . . . . . . . . . . . . . . . . . . . . . 269 6.2.3. Flow of compressible inviscid fluid. . . . . . . . . . . . . . . . . . . 270 6.2.4. Nature of equations in inviscid flows. . . . . . . . . . . . . . . . . . 271 6.2.5. Elementary solutions in irrotational flows . . . . . . . . . . . . . . . 273 6.2.6. Surface waves in shallow water . . . . . . . . . . . . . . . . . . . . . 284

6.3. Orders of magnitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288 6.3.1. Introduction and discussion of a simple example. . . . . . . . . . . 288 6.3.2. Obtaining approximate values of a solution . . . . . . . . . . . . . . 291

6.4. Small parameters and perturbation phenomena. . . . . . . . . . . . . . . 296 6.4.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296 6.4.2. Regular perturbation. . . . . . . . . . . . . . . . . . . . . . . . . . . . 296 6.4.3. Singular perturbations. . . . . . . . . . . . . . . . . . . . . . . . . . . 305

6.5. Quasi-1D flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309 6.5.1. General properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309 6.5.2. Flows in pipes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314 6.5.3. The boundary layer in steady flow . . . . . . . . . . . . . . . . . . . 319

6.6. Unsteady flows and steady flows . . . . . . . . . . . . . . . . . . . . . . . 327 6.6.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327 6.6.2. The existence of steady flows . . . . . . . . . . . . . . . . . . . . . . 328 6.6.3. Transitional regime and permanent solution. . . . . . . . . . . . . . 330 6.6.4. Non-existence of a steady solution . . . . . . . . . . . . . . . . . . . 334

Table of Contents ix

Chapter 7. Measurement, Representation and Analysis of Temporal Signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339

7.1. Introduction and position of the problem . . . . . . . . . . . . . . . . . . 339 7.2. Measurement and experimental data in flows. . . . . . . . . . . . . . . . 340

7.2.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 340 7.2.2. Measurement of pressure . . . . . . . . . . . . . . . . . . . . . . . . . 341 7.2.3. Anemometric measurements . . . . . . . . . . . . . . . . . . . . . . . 342 7.2.4. Temperature measurements . . . . . . . . . . . . . . . . . . . . . . . 346 7.2.5. Measurements of concentration . . . . . . . . . . . . . . . . . . . . . 347 7.2.6. Fields of quantities and global measurements. . . . . . . . . . . . . 347 7.2.7. Errors and uncertainties of measurements . . . . . . . . . . . . . . . 351

7.3. Representation of signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357 7.3.1. Objectives of continuous signal representation . . . . . . . . . . . . 357 7.3.2. Analytical representation . . . . . . . . . . . . . . . . . . . . . . . . . 360 7.3.3. Signal decomposition on the basis of functions; series and elementary solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361 7.3.4. Integral transforms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363 7.3.5. Time-frequency (or timescale) representations . . . . . . . . . . . . 374 7.3.6. Discretized signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381 7.3.7. Data compression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385

7.4. Choice of representation and obtaining pertinent information . . . . . . 389 7.4.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389 7.4.2. An example: analysis of sound . . . . . . . . . . . . . . . . . . . . . 390 7.4.3. Analysis of musical signals . . . . . . . . . . . . . . . . . . . . . . . 393 7.4.4. Signal analysis in aero-energetics . . . . . . . . . . . . . . . . . . . . 402

Chapter 8. Thermal Systems and Models . . . . . . . . . . . . . . . . . . . . . 405

8.1. Overview of models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405 8.1.1. Introduction and definitions . . . . . . . . . . . . . . . . . . . . . . . 405 8.1.2. Modeling by state representation and choice of variables . . . . . . 408 8.1.3. External representation . . . . . . . . . . . . . . . . . . . . . . . . . . 410 8.1.4. Command models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411

8.2. Thermodynamics and state representation. . . . . . . . . . . . . . . . . . 412 8.2.1. General principles of modeling . . . . . . . . . . . . . . . . . . . . . 412 8.2.2. Linear time-invariant system (LTIS) . . . . . . . . . . . . . . . . . . 420

8.3. Modeling linear invariant thermal systems . . . . . . . . . . . . . . . . . 422 8.3.1. Modeling discrete systems . . . . . . . . . . . . . . . . . . . . . . . . 422 8.3.2. Thermal models in continuous media. . . . . . . . . . . . . . . . . . 431

8.4. External representation of linear invariant systems . . . . . . . . . . . . 446 8.4.1. Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 446 8.4.2. External description of linear invariant systems . . . . . . . . . . . 446

8.5. Parametric models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 451 8.5.1. Definition of model parameters . . . . . . . . . . . . . . . . . . . . . 451 8.5.2. Established regimes of linear invariant systems . . . . . . . . . . . 453

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8.5.3. Established regimes in continuous media . . . . . . . . . . . . . . . 458 8.6. Model reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465

8.6.1. Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465 8.6.2. Model reduction of discrete systems . . . . . . . . . . . . . . . . . . 466

8.7. Application in fluid mechanics and transfer in flows . . . . . . . . . . . 474

Appendix 1. Laplace Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . 477

A1.1. Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 477 A1.2. Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 477 A1.3. Some Laplace transforms . . . . . . . . . . . . . . . . . . . . . . . . . . 478 A1.4. Application to the solution of constant coefficient differential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 479

Appendix 2. Hilbert Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . 481

Appendix 3. Cepstral Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483

A3.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483 A3.2. Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483 A3.3. Example of echo suppression . . . . . . . . . . . . . . . . . . . . . . . . 484 A3.4. General case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485

Appendix 4. Eigenfunctions of an Operator . . . . . . . . . . . . . . . . . . . . 487

A4.1. Eigenfunctions of an operator . . . . . . . . . . . . . . . . . . . . . . . . 487 A4.2. Self-adjoint operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 487

A4.2.1. Eigenfunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 487 A4.2.2. Expression of a function of f using an eigenfunction basis-set . . 488

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 489

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 497

Preface

The study of fluid mechanics and transfer phenomena in flows involves the association of difficulties which are encountered in different disciplines: thermodynamics, mechanics, thermal conduction, diffusion, chemical reactions, etc. This book is not intended to be an encyclopaedia, and we will thus not endeavour to cover all of the aforementioned disciplines in a detailed fashion. The main objective of the text is to present the study of the movement of fluids and the main consequences in terms of the transfer of mass and heat. The book is the result of many years of teaching and research, both theoretical and applied, in scientific domains which are often considered separately. In effect, the development of new disciplines which are at the same time specialized and universal was very much a characteristic of science in the 20th century. Thus, signal processing, system analysis, numerical analysis, etc. are all autonomous disciplines and indispensable means for students, engineers or researchers working in the domain of fluid mechanics and energetics. In the same way, various domains such as the design of chemical reactors, the study of the stars and meteorology require a solid knowledge of fluid mechanics in addition to that of their specific topics.

This book is primarily aimed at students, engineers and researchers in fluid mechanics and energetics. However, we feel that it can be useful for people working in other disciplines, even if the reading of some of the more theoretical and specialized chapters may be dispensable in this case. The science and technology of the first half of the 20th century was heavily rooted in classical mechanics, with concepts and methods which relied on algebra and differential and integral calculus, these terms being taken into account in the sense they were used at that time. Furthermore, scientific thought was fundamentally deterministic during this period, even if the existence of games of chance using mechanical devices (dice, roulette, etc.) seemed far from the philosophy of science or Cauchys theorem. Each time has

xii Fundamentals of Fluid Mechanics and Transport Phenomena

its concepts, which are based on the current state of knowledge, and the science of fluid mechanics was reduced for the most part to semi-empirical engineering formulae and to particular analytical solutions. Between the 1920s and the 1950s, our ideas on boundary layers and hydrodynamic stability were progressively elucidated. Studies of turbulence, which began in the 1920s from a conceptual statistical point of view, have really only made further progress in the 1970s, with the writing of the balance equations using turbulence models with a physical basis. This progress remains quite modest, however, considering the immensity of the task which remains.

It should be noted that certain disciplines have seen a spectacular renewal since the 1970s for two main reasons: on the one hand, the development of information technology has provided formidable computation and experimental methods, and on the other hand, multidisciplinary problems have arisen from industrial necessities. Acoustics is a typical example: many problems of propagation had been solved in the 1950s-1960s and those which were not made only very slow progress. Physics focused on other fundamental, more promising sectors (semiconductors, properties of matter, etc.). However, in the face of a need to provide practical solutions to industrial problems (sound generated by fluid flow, the development of ultra-sound equipment, etc.), acoustics became an engineering science in the 1970s. Acoustics is indeed a domain of compressible fluid mechanics and it will constitute an integral part of our treatment of the subject.

Parallel to this, systems became an object of study in themselves (automatic control) and the possibilities of study and understanding of the complexity progressed (signal processing, modeling of systems with large numbers of variables, etc.). Determinism itself is now seen in a more modest light: it suffices to remember the variable level of our ambitions with regard to meteorological prediction in the last 30 years to see that we have not yet arrived at a point where we have a definite set of concepts. Meteorological phenomena are largely governed by fluid mechanics.

The conception of this book results from the preceding observations. The author refuses to get into the argument which consists of saying that the time of analytical solutions has passed and that numerical simulation will solve all our problems. The reality is clearly more subtle than this: analytical solution in the broad sense, that is, the obtaining of results derived from reasoning and mathematical concepts, is the basis of physical concepts. Computations performed by computers by themselves cannot provide any more insight than an experiment, although both must be performed with great care. The state of knowledge and of understanding of mechanisms varies depending on the domain studied. In particular, the science of turbulence is still at a somewhat embryonic stage, and the mystery of turbulent solutions of the Navier-Stokes equations is far from being thoroughly cleared up.

Preface xiii

We are still at the stage of Galileo who attempted to understand mechanics without the ideas of differential calculus. Nobody can today say precisely what are the difficulties to be solved, and the time which will be required for their resolution (10 years, a century or 10 centuries). We will therefore present the state of our knowledge in the current scientific context by also considering some of the accompanying disciplines (thermodynamics, ideas related to partial differential equations, signal processing, system analysis) which are directly useful to the concepts, modeling, experiments and applications in fluid mechanics and energetics of flows. We will not cover specific combustion phenomena, limiting ourselves to a few simplified cases of physico-chemical reactions.

This book covers the necessary fundamentals for the study and understanding of the specific concepts and general properties of flows: the establishment and discussion of the balance equations of extensive quantities in fluid motions, the transport of these quantities by convection, wave-propagation or diffusion. These physical concepts are issued from the comprehension of theoretical notions associated with equations, such as characteristic curves or surfaces, perturbation methods, modal developments (Fourier series, etc.) and integral transforms, model reduction, etc. These mathematical aspects are either consequences of properties of partial differential equations or derived from other disciplines such as signal processing and system analysis, whose impact is important in every scientific or technological domain. They are discussed and illustrated by some elementary problems of fluid mechanics and thermal conduction, including measurement methods and experimental data processing This book is an introduction to the study of more specialized topics of fluid flow and transfer phenomena encountered in different domains of application: incompressible or compressible flow, dynamic and thermal boundary layers, natural or mixed convection, 3D boundary layers, physico-chemical reactions in flows, acoustics in flows, aerodynamic sound, thermoacoustics, etc.

Chapter 1 is devoted to a synthetic presentation of thermodynamics. After recalling the basics of the representation of material systems, thermostatics is covered in an axiomatic fashion which avoids the use of differential formulations and which allows for a simplified presentation of classical results. Taking entropy dynamics as a starting point, the thermodynamics of non-equilibrium states is then discussed using simple examples with phenomenological laws of linear thermodynamics.

The continuous medium at rest is obtained by taking the limit of discrete systems in Chapter 2. The exchange of extensive quantities is modeled by means of flux densities, and irreversible thermodynamics leads to the diffusion equations. Some reminders of fluid statics are given. We then discuss the difficulties specific to the diffusion of matter.