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

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Unformatted text preview: Lectures in Turbulence for the 21st Century William K. George Professor of Turbulence Chalmers University of Technology Gothenburg, Sweden 2 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 4 CONTENTS 3.5 The importance of non-linearity . . . . . . . . . . . . . . . . . . . 44 3.6 The Eddy Viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.7 The Reynolds Stress Equations . . . . . . . . . . . . . . . . . . . 51 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 ofity over the world around us....
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This note was uploaded on 09/16/2011 for the course ME 563 taught by Professor Staff during the Spring '11 term at Auburn University.

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

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