Continuum Methods of Physical Modeling ONLINE LIBRARY Physics and Astronomy springeronline.com Springer-Verlag Berlin Heidelberg GmbH Kolumban Hutter Klaus Johnk Continuum Methods of Physical Modeling Continuum Mechanics, Dimensional Analysis, Turbulence With 61 Figures, 14 Tables, 113 Exercises and Solutions Springer Professor Kolumban Hutter, Ph.D. Dr. Klaus D. Johnk Technische Universitiit Darmstadt FNWIIIBED Institut fiir Mechanik University of Amsterdam Hochschulstrasse 1 Nieuwe Achtergracht 127 64289 Darmstadt, Germany 1018WS Amsterdam, The Netherlands The cover pictures: Laboratory avalanche simulation with a mixture of sand and gravel, at the Depart ment of Mechanics, Darmstadt University of Technology, Germany and a powder snow avalanche in the Nepalese Himalaya, (Photo F. TsCHIRKY, courtesy of Swiss Federal Institute of Snow and Avalanche Research, Davos, Switzerland). Cataloging-in-Publication Data applied for Bibliographic information published by Die Deutsche Bibliothek Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed biblio graphic data is available in the Internet at <http://dnb.ddb.de>. ISBN 978-3-642-05831-8 ISBN 978-3-662-06402-3 (eBook) DOI 10.1007/978-3-662-06402-3 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broad casting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag Berlin Heidelberg GmbH. Violations are liable for prosecution under the German Copyright Law. springeronline.com ©by Springer-Verlag Berlin Heidelberg 2004 Originally published by Springer-Verlag Berlin Heidelberg New York in 2004 Soft cover reprint of the hardcover 1st edition 2004 The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant pro tective laws and regulations and therefore free for general use. Typesetting by the authors using a Springer TE,X macro package Cover design: Erich Kirchner, Heidelberg Printed on acid-free paper 55/3141/tr 5 4 3 2 1 o Preface This book is a considerable outgrowth of lecture notes on Mechanics of en vironmentally related systems I, which I hold since more than ten years in the Department of Mechanics at the Darmstadt University of Technology for upper level students majoring in mechanics, mathematics, physics and the classical engineering sciences. These lectures form a canon of courses over three semesters in which I present the foundations of continuum physics (first semester), those of physical oceanography and limnology (second semester) and those of soil, snow and ice physics in the geophysical context (third semester). The intention is to build an understanding of the mathemati cal foundations of the mentioned geophysical research fields combined with a corresponding understanding of the regional, but equally also the global, processes that govern the climate dynamics of our globe. The present book contains the material (and extensions of it) of the first semester; it gives an introduction into continuum thermomechanics, the methods of dimensional analysis and turbulence modeling. All these themes belong today to the every day working methods of not only environmental physicists but equally also those engineers, who are confronted with continuous systems of solid and fluid mechanics, soil mechanics and generally the mechanics and thermody namics of heterogeneous systems. The book addresses a broad spectrum of researchers, both at Universities and Research Laboratories who wish to fa miliarize themselves with the methods of "rational" continuum physics, and students from engineering and classical continuum physics. Why, however, the threefold division in continuum thermodynamics, di mensional analysis and turbulence modeling? There are several reasons to this end. First, turbulence theory today is part of the working methods of every fluid dynamicist, especially in the geophysical context, such as meteorology, oceanography, limnology, not to mention all the technical applications in en vironmental and mechanical engineering. Second, turbulence research has, in the last twenty years perfected its theoretical formulation to such an extent, that one may well try to present some of its aspects from a viewpoint of general continuum mechanics. Third, it has become apparent in the past few years that for those aspects we are interested in, continuum thermodynam ics possesses the right underlying structure to treat turbulence modeling in a particularly systematic fashion. In other words, one may base the constitutive VI Preface theory of continuous materials on essentially the same, or at least very simi lar, concepts as the formulation of closure conditions in turbulence modeling. To my knowledge, such an approach has not been presented so far in book form. The advantage of such an approach is, however, a considerable increase of the transparency of the material to be learned, a clearer enlightenment into the concepts (which are indeed not very far apart) and, probably most effectively, a reduction of the amount of the topics that are new and must be absorbed for the first time. I emphasize here my opinion that turbulence modeling can profit from an access by continuum mechanics. Which role is now played by dimensional analysis in this context? For one, modeling in turbulence theory, especially when closure conditions must be postulated, depends to a large degree upon simple concepts of dimen sional analysis. Admittedly, this could well be presented without the explicit development of the BUCKINGHAM theorem. However, a clear and relatively rigorous presentation of the methods of dimensional analysis surely facili tates the basic understanding. Moreover, it is a simple fact that, because of the dimensional homogeneity of all equations in mathematical physics, a first bold understanding of a physical problem is gained with the aid of methods of dimensional analysis. At last this same statement also holds for rational continuum mechanics and has always been emphasized by its founder C.A. TRUESDELL. I concur, and this is why we give here a brief and incomplete introduction into this fascinating field of mathematical physics. Dimensional analysis precedes turbulence modeling in this book, because the former is used for the latter much more than vice versa. A word about the role of thermodynamics seems equally to be in order. Today's researchers in rational continuum thermodynamics largely use the CLAUSIUS-DUHEM inequality and the COLEMAN-NOLL approach in deduc ing results in the particular research they are pursuing. In this book this ap proach towards the second law of thermodynamics will also be explained 1, but in a number of applications the more general entropy principle of MULLER will be used2. In so doing it will become apparent that the CLAUSIUS-DUHEM in equality, paired with the COLEMAN-NOLL approach of its exploitation would have been too restrictive in those cases and erroneous results would have been obtained. In this regard this book goes beyond most of the classical treat ments of rational thermodynamics of the last two to three decades. The intention of this book is, apart from presenting its treated subjects, a clear and (somewhat) rigorous mathematical presentation of them on the basis of limited knowledge as a prerequisite. Calculus or analysis of func tions of a single and several variables, linear algebra and (only) the basics of ordinary and partial differential equations are assumed to be known (or having been learned once). Those subjects roughly form the mathematical tool which engineers in Germany learn during the first two years of their 1 See e.g. C. A. TRUESDELL [243). 2 I. MULLER (165). Preface VII university education. In the American system, senior undergraduate or first year graduate-education level is about the background needed to follow the material in this book. On the side of physics knowledge of strength of materi als and dynamics or analytical dynamics courses or a basic course in classical physics should suffice to be able to follow the presented concepts. Even though it is hoped that the book will also be used as a source book by researchers in the broad field of continuum physics, its intention is essentially to form a basis for teaching (and even more so learning). Great care has therefore been devoted in each chapter to formulate a number of exercises, and solutions are given in detail to most of these. The latter is justified for the following reasons: Often, the problems formulated in the exercises constitute complementary material to that presented in the main text of the respective chapters. Occasionally, a thought in a derivation of a certain fact is only briefly touched upon in the main text and the reader is asked to work out the details by himself/herself. At an other time a fact that is needed in the development of the material is only mentioned in the main text, and the reader is asked to corroborate the statement as an exercise. And, finally, additional material that could also be treated in the main text is explained in the exercises as an individual problem. In all these cases knowledge of the material dealt with in the exercises is assumed to be known in later chapters. This is also the reason why solutions to the stated problems are nearly completely outlined. A natural fringe benefit for the reader is obviously the fact of a self-control in his attempts to solve the problems. Most problems were stated for and solved in recitation hours with the students; considerable input has thereby been given by the students for which we express our sincere thanks. The book has been drafted (first) in the German language jointly by both authors. First versions of Chaps. 1 to 6 were written from lecture notes of K. HUTTER by K. JOHNK, when the second author was a postdoctoral assistant of the former. Chap. 7 and the two chapters of Part II: Dimensional Analysis were exclusively written by K. HUTTER. Rough drafts of Part III: Thrbulence, i.e., Chaps. 10, 11 and 13 were written by K. JOHNK, whilst Chap. 12 is due to K. HUTTER. Most problem formulations and their solutions are due to K. HuTTER, but there is a number which are due to K. JOHNK. Because of different profes sional assignments of K. JoHNK since November 1997, the two authors were locally separated. This, together with K. JOHNK more industrious profes sional involvement, made a collaboration as a consequence virtually impossi ble. For this reason, the homogenisation of the entire manuscript, the careful testing, reading and again reading, the dotting of all the i's, the incorporation of the References and the Index are all due to K. HUTTER. Consequently, even though we are both joint authors of the book, K. HUTTER is the sole author responsible for all the errors which still remain. He is particularly VIII Preface thankful to all the readers who would point out to him where they arise. A simple note by e-mail: [email protected] will suffice. The English version now presented was drafted by K. HUTTER with the help of Dr. D. RAJ BARAL. Beside this invaluable help we also were assisted by my secretaries Mrs. R. DANNER and R. RUTSCHER and assistants H. Hi.i'TTEMANN, A. DIENG, S. KTITAREVA, Y.-CH. TAl, E. VASSILIEVA, Y. WANG and A. WILLUWEIT, to all of whom we express our sincere thanks. Before I finish this Preface let me state that writing a book can never be finished, a book has to be abandoned! This I am now going to do, well knowing that it bears its weaknesses, that I would now know how to do it better and being well aware that while pushing this project through all its stages needed isolation and separation from the beloved family members, who all deserve my deepest gratitude. Darmstadt, Autumn 2003 Kolumban Hutter Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Part I. Continuum Mechanics 1. Basic Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.2 Basic Concepts, Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 1.3 Deformation Gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 1.3.1 Definition of the Deformation Gradient. . . . . . . . . . . . . . 23 1.3.2 'fransformation of Surface and Volume Elements . . . . . 24 1.4 Velocity, Acceleration and Velocity Gradient . . . . . . . . . . . . . . . 25 1.5 Deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 1.5.1 Polar Decomposition of the Deformation Gradient . . . . 27 1.5.2 Strain Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 1.5.3 Eigenvalues, Invariants and CAYLEY-HAMILTON Theorem of Tensors of the Second Rank............................... 31 1.5.4 Geometric Linearization . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 1.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 1. 7 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 2. Balance Equations........................................ 51 2.1 General Balance Statements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 2.1.1 Integral Form of the Balance Statements . . . . . . . . . . . . 51 2.1.2 CAUCHY Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 2.1.3 Synopsis of General Balance Statements . . . . . . . . . . . . . 58 2.2 Local Balance Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 2.2.1 REYNOLDS 'fransport Theorem . . . . . . . . . . . . . . . . . . . . 59 2.2.2 Local Balance Equations in the LAGRANGE Representation . . . . . . . . . . . . . . . . . . 61 2.2.3 Local Balance Equations in the EULER Representation 62 2.3 Special Balance Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 2.3.1 Mass Balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64