AN INTRODUCTION TO HYDRODYNAMICS AND WATER WAVES

AN INTRODUCTION TO HYDRODYNAMICS AND WATER WAVES

HYDRODYNAMICA SIVE DE VIRIBUS ET MOTIBUS FLUIDORUM COMMENTARII 'Remember, when discoursing about water, to induce first experience, then reason.' - Leonardo da Vinci

AN INTRODUCTION TO HYDRODYNAMICS AND WATER WAVES Bernard Le Mehaute SPRINGER SCIENCE+ BUSINESS MEDIA, LLC 1976

Bernard Le Mehaute Senior Vice President Tetra Tech, Inc. 630 North Rosemead Boulevard Pasadena, California 91107 This volume has been totally revised from an earlier version published by the U.S. Government Printing Office as ESSA Technical Reports ERL 118-POL 3-1 and 3-2. 1976 by Springer Science+Business Media New York Originally published by Springer-Verlag New York in 1976 Softcover reprint of the hardcover 1st edition 197 6 All rights reserved. No part of this book may be translated or reproduced in any form without written permission from Springer Science+Business Media, LLC. Library of Congress Cataloging in Publication Data Le Mehaute, Bernard, 1927- An introduction to hydrodynamics and water waves. Includes bibliographical references and index. I. Hydrodynamics. 2. Water waves. I. Title. QA9ll.L39 532'.5 75-12754 ISBN 978-3-642-85569-6 DOI 10.1007/978-3-642-85567-2 ISBN 978-3-642-85567-2 (ebook) Additional material to this book can be downloaded from http://extras.springer.com

Preface Hydrodynamics is the science which deals with the motion of liquid in the macroscopic sense. It is essentially a field which is regarded as applied mathematics because it deals with the mathematical treatments of basic equations for a fluid continuum obtained on a purely Newtonian basis. It is also the foundation of hydraulics, which, as an art, has to compromise with the rigorous mathematical treatments because of nonlinear effects, inherent instability, turbulence, and the complexity of" boundary conditions" encountered in engineering practice. Therefore, this book can be considered as the text for a course in basic hydrodynamics, as well as for a course in the fundamentals of hydraulic and related engineering disciplines. In the first case, the students learn how to make use of their mathematical knowledge in a field of physics particularly suitable to mathematical treatments. Since they may have some difficulty in representing a physical phenomenon by a mathematical model, a great emphasis has been given to the physical concepts of hydrodynamics. For students with an undergraduate training in engineering, the difficulty may be a lack of appropriate mathematical tools. Their first contact with hydraulics has been on an essentially practical basis. They may be discouraged in attempting the study of such books as Hydrodynamics, by Lamb, which remains the bible of hydrodynamicists. Hence, mathematical intricacies have been introduced slowly and progressively. Also, the emphasis on the physical approach has made it possible to avoid mathematical abstractions so that a concrete support may be given to equations. Finally, the author has tried to make this book selfcontained in the sense that a practicing engineer who wants to improve his theoretical background can study hydrodynamics by himself without attending lectures. Too often articles in scientific journals present some discouraging aspects to practicing engineers and the most valuable messages can only reach a few specialists. It is felt that the learning of some basic theories will help hydraulic engineers to keep abreast of and participate in new developments proposed by theorists. Considering that a good assimilation of the basis is essential before further study, great care has been taken to v

vi develop a clear understanding, both mathematically and physically, of the fundamental concepts of theoretical hydraulics. The introduction of mathematical simplifications and assumptions, often based on physical considerations, has also been developed by examples. The mathematical difficulties have been cleared up by introducing them progressively and by developing all the intermediate calculations. Also, all the abstract concepts of theoretical hydraulics have been explained as concretely as possible by use of examples. It will appear that the first chapter is the easiest to understand, and it is assumed that the mathematical background increases as the student progresses toward the end of the book. However, it is taken for granted that the student already has some notion of elementary hydraulics. Finally, the succession of the various chapters have been chosen in order to build up a structure as logical and as deductive as possible in order to avoid that the various subjects appear as a succession of different mathematical recipes rather than as a unique and logical subject. Part One deals with the establishment of the fundamental differential equations governing the flow motion in all possible cases. The possible approximations are also indicated. Part Two deals with general methods of integrations and the mathematical treatments of these equations. Integrations of general interest, and integrations in some typical particular cases are presented. Part Three is devoted to water wave theories, as one of the most important topics of hydrodynamics. It is pointed out that the emphasis of the book is on water waves. Therefore the treatment of motion of compressible fluids has been judged beyond the scope of this book, with a few exceptions. Also, almost all the calculations are presented in a Cartesian (or cylindrical) system of coordinates. Vectorial and tensorial operations have been minimized in order to reduce the necessary mathematical background. However, vectorial and tensorial notations are slowly introduced for sake of recognition in the literature. It is hoped that this book will entice students gifted in mathematics to apply their capabilities to the study of fluid motion and dynamical oceanography. It is hoped also that it will instill in engineering students the desire for further study in hydrodynamics and mathematics. It is also hoped that the book will be of great help to students in hydraulics, civil and coastal engineering, naval architecture, as well as in physical oceanography, marine geology, and sedimentology, who want to learn or revise one of the theoretical aspect of their future profession. ACKNOWLEDGMENTS The author wishes to express his deep gratitude to Nicholas Boratynski, President of Tetra Tech, Inc. Without his encouragement and support, these lecture notes would not have been revised and published. Many valuable suggestions and contributions are credited to the senior engineering and scientific staff of Tetra Tech. The author would also like to acknowledge Dr. Viviane Rupert of the Lawrence Livermore Laboratory for her help in editing the book. Bernard Le Mehaute Pasadena, California

Contents PART ONE vii Establishing the Basic Equations that Govern Flow Motion 1. Basic Concepts and Principles 3 2. Motions of Fluid Elements; Rotational and Irrotational Flow 16 3. The Continuity Principle 31 4. Inertia Forces 37 5. Applied Forces 51 6. Forms of the Momentum Equation: Equations of Euler and Navier-Stokes- 59 7. Turbulence: Mean and Fluctuating Components of Motion 71 8. Turbulence Effects: Modern Theories 80 9. Flow in a Porous Medium: Law of Darcy 89 PART TWO Some Mathematical Treatments of the Basic Equations 10. The Bernoulli Equation 101 11. Flow Pattern, Stream Function, Potential Function 116 12. The Momentum Theorem and Its Applications 137 13. The Boundary Layer, Flow in Pipes, Drag and Added Mass 154 14. Open-Channel Hydraulics 182 PART THREE Water Wave Theories 15. An Introduction to Water Waves 197 16. Linear Small Amplitude Wave Theories 212

Contents vm 17. Finite Amplitude Waves 239 18. The Long Wave Theory 256 Appendix A: Wave Motion as a Random Process 273 Appendix B: Similitude and Scale Model Technology 286 Notation 301 Answers to Selected Problems 305 Index 317