STATISTICAL FLUID MECHANICS: Mechanics of Turbulence Volume 1 Monin and M. Yaglom A. S. A. English edition updated, augmented and revised by the authors Edited by L. Lumley John THE MIT PRESS Cambridge, Massachusetts, and London, England Originaplulbyl isihne1d 96 5 byN aukaP resMso,s cowu,n detrh e title Statisicheskaya gidromekhanika -Mekhanika Turbulentnosti, byA ndreSy. M onina ndA kivMa. Yaglom. Translafrtoemd t heR ussiabnyS cripTteac hnicIan,c . Seconpdr intiDnegc,e mbe1r97 3 1977 Third printing, March printing,1 979 Fourth May Englitsrha nslactoipoynr igh1t9©71 by TheM assachusIentsttsi toufTt eec hnology Allr ighrtess ervNeod p.a rotf t hibso okm ayb er eproduicne d anyfo rm orb ya nym eanse,l ectroonrmi ecc hanicianlc,l uding photocopyriencgo,r dionrbg y,a nyi nformatisotno raagned retriesvyaslt ewmi,t hopuetr missiinow nr itifrnogm thep ublisher. ISBN0 262 13062 9 (hardcover) LibraorfyC ongrecsast alcoagr ndu mber7:0- 110232 Printiendt heU niteSdt atoefsA merica AUTHORS' PREFACE TO THE ENGLISH EDITION The flows of fluids with which one has to deal in engineering, and which one meets in nature, are turbulent in the overwhelming majority of cases, and their description demands a statistical approach. Laminar flows, which are quite accessible to individual descriptions, occur with exotic infrequency. We are persuaded that fluid mechanics cannot be limited to the study of these seldom encountered special cases, and that the classical description of individual laminar flows, for all its unquestioned importance and value, must be considered only as an introductory chapter to the theory of real turbulent flows, in which the objects of investigation are the properties of ensembles of flows, arising from macroscop­ ically identical external conditions. We would like such a view to spread through the entire fluid mychanics community, and this is one of the goals of our book. For this reason we are very happy with the appearance of the English edition, which will help spread our views on fluid mechanics in English-speaking countries. We have tried to summarize in this book the majority of the fundamental works and ideas of modern turbulence theory, de­ scribed and discussed in studies from various countries. This is why the book has turned out to be so thick. But, of course, our own scientific interests must be reflected in the selection of material, and considerable attention has been given to problems on which we ourselves have done substantial work-particularly the theory of the local structure of developed turbulence due to A. N. Kolmogorov, whose students we were, and the theory of turbulence in stratified media, which finds wide applications, primarily in geophysics. The mechanics of turb\Jlence is a lively, actively developing science, and after the Russian edition of our book saw the light (Volume l in 1965 and Volume 2 in 1967), a large number of new iii AUTHORS' PREFACE iv a r in this area. The present edition has ea e abneden i nteresetinisgh ewdo rkwsi thp pinfordm ation on many new works (the e r pl n included in the book has, as a result, grown by several bhiubnlidorgerda pehnyt ries) and by the inclusion of a series of improvements and additions (all of this work was carried out by the second author). Particularly great changes were undergone by Section 2, dealing with hydrodynamic stability and transition to turbulence, and Section 8, in which are set forth experimental results on turbulence in a thermally stratified boundary layer. We consider it our pleasure to express our deep gratitude to our editor, Prof. L. Lumley, who did a great deal to improve the English editionJ .o f our book. A. Monin A. SM. . Yaglom EDITOR'S PREFACE TO THE ENGLISH EDITION This is a translation of the first Russian edition of In editing the manuscript of the traCnrslnartniocrnn, 4eexctKeanH­ srtivtJ:te puosMe cwxaasH mHKaad. e of the translation of Volume 1 prepared by the Joint Publications Research Service at the behest of K.L. Calder of the Environmental Science Services Administration, and of the tranUs.lSa­. tion of Chapter 4, Volume 1, prepared at the instigation of F. Pasquill of the British Meteorology Research Division. The cooperation of both these individuals is gratefully acknowledged. The edited English manuscript was sent to Yaglom, who made very extensive additions, corrections and revisions to the technical content, after which it was again edited. Russian technical style tends to be turgid with internal cross-references which sound redundant to an English ear; in addition, the inflected character of the language makes possible a complexity of sentence structure that is dizzying. A conscientious translator hesitates to paraphrase too freely, with the consequence that the translation retains an unmistakable flavor of the original. Yaglom has an excellent ear for English style, and as he revised he also made innumerable suggestions for freer paraphrases, which sound more natural in English. As editor, I wish to express my gratitude for his cooperation, giving him full credit for what is good in this translation, while taking the blame for remaining inade­ quacies. L. Lumley J. v FOREWORD The theory of turbulence discussed in this book is based on the usual macroscopic description of flows of liquids and gases, considered as continua, and on the classical equations of fluid mechanics. However, unlike ordinary fluid mechanics, the theory of turbulence does not study individual fluid flows but the statistical properties of an ensemble of flows having macroscopically identical external condi­ tions. Hence, the title of this book, Statistical Fluid Mechanics. However, we must stress that questions dealing with the deduction of the macroscopic equations of fluid mechanics from the statistical laws of the kinetic theory of gases, which sometimes are also referred to by this title, are not discussed in the present work. The basic concepts and ideas relating to turbulence and our approach to the presentation of the mechanics of turbulence are outlined in the Introduction, together with a brief sketch of the historical development of the theory of turbulence, and the plan of bo. th volumes of the present work. The Introduction is intended to describe in the most general terms the main problems and methods of the recent theory of turbulence and to explain some of its practical applications. Naturally, we have had to make use here of many concepts which will be expanded in detail only in subsequent parts of the book. The book is divided into chapters, sections and subsections. The numbering of the chapters and sections runs continuously, while the subsections are numbered within each section. The number of the section and of the subsection is indicated in the subsection number separately (e.g., the fourth subsection of the second section is denoted as 2.4 ). The equations are also numbered in a similar manner: e.g., Eq. (2.15) is the fifteenth equation in the second section. When referring to the works listed in the bibliographies at the end of each volume, we cite the name of the author and, in brackets, the date of publication of the work. In cases when several works of a given author are cited which all appeared in the same year, these are denoted further by letters of the alphabe.t . Except . m vii FOREWORD viii the Introduction, the initials of the authors are given only in cases where one must distinguish between two authors with the same surname. We have tried, as far as possible, to use generally accepted notation in this book. In the numerous cases where different authors use different symbols for the same quantities, we usually decided to select one of these, and not to introduce new notation. In several cases this has led to the same symbol being used for different quantities in different parts of the book. Sometimes, for various reasons, it also proved convenient to denote the same quantity by a different symbol in different sections of the book. In all these cases the notation used is specified in the text. This book represents the combined work of both authors. We feel that we must point out the great influence on the writing of it, and on our own work in the field of turbulence, of our frequent discussions with Andrei Nikolaevich Kolmogorov, our Professor in our student days. We have tried to reflect many of Kolmogorov's ideas in this book. We are also greatly indebted to A. M. Obukhov who was one of those who initiated the writing of this book, and with whom we discussed the selection of the material and the details of the exposition of many questions. Some parts of the manuscript were read by L. A. Dikiy, Ye. A. Novikov and V. I. Tatarskiy, who made a number of profitable comments. In discussions on the analysis of the experimental data on atmospheric turbulence, A. S. Gurvich and L. V. Tsvang were active. G. S. Golitsyn helped us in the preparation and editing of the manuscript. To all these colleagues we wish to express our sincere gratitude. A. Monin A. SM. . Vaglom CONTENTS Authors' Preface to the English Edition iii Editor's Preface to the English Edition Foreword vivi Introduction Chapter 1 Laminar and Turbulent Flows 27 1. Equations of Fluid Dynamics and Their Consequences 27 1.1 System Dynamical Equations for an of Incompressible Fluid 27 1.2 Simple Flows of Incompressible Fluid 31 1.3 The Reynolds Number and Similarity Criteria 38 1.4 Flows with Large Reynolds Numbers; The Boundary Layer 41 1.5 General Equation of the Heat Budget and the Thermal Conduction Equation; Forced and Free Convection 52 1.6 Similarity Criteria for a Thermally Inhomogeneous Fluid; The Thermal Boundary Layer 59 I .7 Small Oscillations of Compressible Fluid 64 2. Hydrodynamic Instability and aT ransition to Turbulence 71 2. Concept of Turbulence; Empirical Data on Transition I to Turbulence in Tubes and Boundary Layers 71 2.2 Turbulent Flow Past Solid Bodies; Boundary-layer Separation, the Drag Crisis and the Mechanism of Boundary-layer Transition 79 2.3 Hydrodynamic Instability 85 2.4 Simple Examples of Absolutely Unstable Fluid 2.5 M�atMhe matical Formulation of the Stability Problem � for Infinitesimal Disturbances 92 2 .6 Stability of Flow between Two Rotating Cylinders 96 2.7 Stability of a Layer of Fluid Heated from Below 107 2.8 Stability of Parallel Flows 1 1 1 2 Stability to Finite Disturbances; Growth of .9 Disturbances and Transition to Turbulence 147 ix