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important_george - Lectures in Turbulence for the 21st...

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Lectures in Turbulence for the 21st Century William K. George Professor of Turbulence Chalmers University of Technology Gothenburg, Sweden
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Contents 1 The Nature of Turbulence 5 1.1 The turbulent world around us . . . . . . . . . . . . . . . . . . . 5 1.2 What is turbulence? . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.3 Why study turbulence? . . . . . . . . . . . . . . . . . . . . . . . . 9 1.4 The cost of our ignorance . . . . . . . . . . . . . . . . . . . . . . 10 1.5 What do we really know for sure? . . . . . . . . . . . . . . . . . . 11 1.6 Our personal adventure . . . . . . . . . . . . . . . . . . . . . . . . 12 1.7 A brief outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2 The Elements of Statistical Analysis 15 2.1 Foreword . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.2 The Ensemble and Ensemble Averages . . . . . . . . . . . . . . . 16 2.2.1 The mean (or ensemble) average . . . . . . . . . . . . . . 16 2.2.2 Fluctuations about the mean . . . . . . . . . . . . . . . . 18 2.2.3 Higher moments . . . . . . . . . . . . . . . . . . . . . . . 19 2.3 Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.3.1 The histogram and probability density function . . . . . . 20 2.3.2 The probability distribution . . . . . . . . . . . . . . . . . 22 2.3.3 Gaussian (or normal) distributions . . . . . . . . . . . . . 22 2.3.4 Skewness and kurtosis . . . . . . . . . . . . . . . . . . . . 24 2.4 Multivariate Random Variables . . . . . . . . . . . . . . . . . . . 25 2.4.1 Joint pdfs and joint moments . . . . . . . . . . . . . . . . 25 2.4.2 The bi-variate normal (or Gaussian) distribution . . . . . . 28 2.4.3 Statistical independence and lack of correlation . . . . . . 30 2.5 Estimation from a Finite Number of Realizations . . . . . . . . . 31 2.5.1 Estimators for averaged quantities . . . . . . . . . . . . . . 31 2.5.2 Bias and convergence of estimators . . . . . . . . . . . . . 31 2.6 Generalization to the estimator of any quantity . . . . . . . . . . 34 3 Reynolds Averaged Equations 37 3.1 The Equations Governing the Instantaneous Fluid Motions . . . 37 3.2 Equations for the Average Velocity . . . . . . . . . . . . . . . . . 39 3.3 The Turbulence Problem . . . . . . . . . . . . . . . . . . . . . . . 41 3.4 The Origins of Turbulence . . . . . . . . . . . . . . . . . . . . . . 42 3
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4 CONTENTS 3.5 The importance of non-linearity . . . . . . . . . . . . . . . . . . . 44 3.6 The Eddy Viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.7 The Reynolds Stress Equations . . . . . . . . . . . . . . . . . . . 51
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Chapter 1 The Nature of Turbulence 1.1 The turbulent world around us The turbulent motion of fluids has captured the fancy of observers of nature for most of recorded history. From howling winds to swollen floodwaters, the om- nipresence of turbulence paralyzes continents and challenges our quest for author- ity over the world around us. But it also delights us with its unending variety of artistic forms. Subconsciously we find ourselves observing exhaust jets on a frosty day; we are willingly hypnotized by licking flames in an open hearth. Babbling brooks and billowing clouds fascinate adult and child alike. From falling leaves to the swirls of cream in steaming coffee, turbulence constantly competes for our attention. Turbulence by its handiwork immeasurably enriches the lives of even those who cannot comprehend its mysteries. Art museums are filled with artists attempts to depict turbulence in the world around us. The classic sketch of Italian renaissance artist and engineer, Leonardo da Vinci, shown in Figure 1.1 represents both art and early science. And as the tongue-in-cheek poem below by Corrsin (one of the turbulence greats of the past century) shows, even for those who try, the distinction between art and research is often difficult to make. SONNET TO TURBULENCE by S. Corrsin 1 (For Hans Liepmann 2 on the occasion of his 70th birthday, with apologies to Bill S. and Liz B.B.) Shall we compare you to a laminar flow?
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