Lectures in Turbulence for the 21st Century

1 Lectures in Turbulence for the 21st CenturyWilliam K. GeorgeDepartment of AeronauticsImperial College of LondonLondon, UKandProfessor of Turbulence EmeritusDepartment of Applied MechanicsChalmers University of TechnologyGothenburg, Sweden16 January 20132 PrefaceThese notes have evolved over many years and are o ered freely tostudents and researchers everywhere. In my own personal history I came fromrather humble beginnings, and was the bene ciary of very generous scholarshipand fellowship support, even with books provided. Without this generosity on thepart of the taxpayers of the State of Maryland, the USA Federal Goverment andthe Johns Hopkins University, I could never be where I am today.

2 So these notesare my gift to all those who come behind, and especially those who nd the costof most books to be a real nancial struggle. If you reference them, please refer tothe website where others can nd them ( ). They willprobably never stop evolving, so it might be a good idea to check once in a whilefor updated notes actually started as book on Turbulence I was writing in the early1980's, and which was nearly nished and ready to be published. I had taughtturbulence classes for more than a decade (at SUNY/Bu alo) and thought I hadfound a way to improve the available texts. Fortunately (or unfortunately forpublication), one day I ended up in class without my notes, and in trying toanswer questions asked to me by the class, I ended up in a very di erent placethan I expected.

3 Some of the results have been published in various places, andsome are referred to herein. But it was clear to me from the very rst momentthat the book had to go: I no longer believed to be true what I was saying in last 30 years have been an interesting journey as successive waves of studentshave asked even more di cult questions, and as I wrestled to gure out what wastrue and not true. I have tried in these notes to provide to the best of my abilityan honest account of Turbulence and what I think we know for sure and what wedo not. I remember despairing as a student that it seemed that all the problemswere solved, and there was really nothing fundamental left to do.

4 I doubt anycareful reader of these notes will come to that erroneous conclusion. The goal wasnot to be comprehensive, so there is much left out. But instead the goal is toinspire continued study, and to provide a foundation for have also tried to make these notes fun to read. If you are expecting a totallyserious account, you will probably be disappointd. I have included my opinionswhen I thought it would help in understanding, and also anedotes when I couldthink of them. The point is that Turbulence (like all research) is done by is probably true that the ones who do the best work are also having the mostfun. I hope the newcomers to the eld nd that do not hesitate to ask me questions or point out places where I havescrewed up.

5 My email address is Whenever possibleI will try to The Nature of The turbulent world around us .. What is Turbulence ? .. Why study Turbulence ? .. The cost of our ignorance .. What do we really know for sure? .. Our personal adventure .. A brief outline .. 172 The Elements of Statistical Foreword .. The Ensemble and Ensemble Averages .. The mean (or ensemble) average .. Fluctuations about the mean .. Higher moments .. Probability .. The histogram and probability density function .. The probability distribution .. Gaussian (or normal) distributions .. Skewness and kurtosis.

6 Multivariate Random Variables .. Joint pdfs and joint moments .. The bi-variate normal (or Gaussian) distribution .. Statistical independence and lack of correlation .. Estimation from a Finite Number of Realizations .. Estimators for averaged quantities .. Bias and convergence of estimators .. Generalization to the estimator of any quantity .. 383 Reynolds Averaged The Equations Governing the Instantaneous Fluid Motions .. Equations for the Average Velocity .. The Turbulence Problem .. The Origins of Turbulence .. The importance of non-linearity .. The Eddy Viscosity .. The Reynolds Stress Equations.

7 564 The Turbulence Kinetic The Kinetic Energy of the Fluctuations .. The Dissipation Rate .. The Production .. The Transport (or Divergence) Terms .. The Intercomponent Transfer of Energy .. 725 A First Look at Homogeneous Introduction .. Why are homogeneous ows important? .. Decaying Turbulence : a brief overview .. A second look at simple shear ow Turbulence .. 886 Turbulent Free Shear Introduction .. The averaged equations .. The shear layer quations .. Order of magnitude estimates .. The streamwise momentum equation .. The transverse momentum equation .. The free shear layer equations.

8 Two-dimensional Turbulent Jets .. Similarity analysis of the plane jet .. Implications of the Reynolds stress equations .. Other Free Shear Flows .. 1167 WALL-BOUNDED TURBULENT Introduction .. Review of laminar boundary layers .. The \outer" turbulent boundary layer .. The \inner" turbulent boundary layer .. The viscous sublayer .. The linear sublayer .. The sublayers of the constant stress region .. Pressure gradient boundary layers and channel ow .. The inertial sublayer .. Some history .. Channel and Pipe Flows .. Boundary Layers .. Summary and conclusions .. 142 CONTENTS58 Stationarity Random Processes statistically stationary in time.

9 The autocorrelation .. The autocorrelation coe cient .. The integral scale .. The temporal Taylor microscale .. Time averages of stationary processes .. Bias and variability of time estimators .. 1579 Homogeneous Random Random elds of space and time .. Multi-point correlations .. Spatial integral and Taylor microscales .. Integral scales .. Taylor microscales .. Symmetries .. Implications of Continuity .. Reduction ofBi;j(~r) to three independent components .. Relations among the derivative moments .. Elimination of the cross-derivative moments .. The `homogeneous' dissipation .. Pressure uctuations in homogeneous Turbulence .

10 Isotropy .. An example from uid mechanics: viscous stress .. Isotropic single-point correlations .. Isotropic two-point statistics .. Derivative moments in isotropic Turbulence .. Isotropic integral scale and Taylor microscales .. Axisymmetric Turbulence .. 18110 Decomposing Scales of Turbulence .. The anatomy of turbulent motions .. Fields of Finite Extent .. Homogeneous Fields .. Are Homogeneous Fields and Periodic Fields the Same? .. Inhomogeneous elds of In nite Extent .. 19411 Decomposing Homogeneous Generalized functions .. Fourier transforms of homogeneous Turbulence .. The Three-dimensional Energy Spectrum Function.