lctr-notes634 - INTRODUCTORY LECTURES on TURBULENCE...

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Unformatted text preview: INTRODUCTORY LECTURES on TURBULENCE Physics, Mathematics and Modeling J. M. McDonough Departments of Mechanical Engineering and Mathematics University of Kentucky c 2004 Contents 1 Fundamental Considerations 1 1.1 Why Study Turbulence? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Some Descriptions of Turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.3 A Brief History of Turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.3.1 General overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.3.2 Three eras of turbulence studies . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.4 Definitions, Mathematical Tools, Basic Concepts . . . . . . . . . . . . . . . . . . . . 12 1.4.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.4.2 Mathematical tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 1.4.3 Further basic concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 1.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 2 Statistical Analysis and Modeling of Turbulence 63 2.1 The Reynolds-Averaged NavierStokes Equations . . . . . . . . . . . . . . . . . . . . 64 2.1.1 Derivation of the RANS equations . . . . . . . . . . . . . . . . . . . . . . . . 64 2.1.2 Time-dependent RANS equations . . . . . . . . . . . . . . . . . . . . . . . . . 67 2.1.3 Importance of vorticity and vortex stretching to turbulence . . . . . . . . . . 69 2.1.4 Some general problems with RANS formulations . . . . . . . . . . . . . . . . 72 2.1.5 Reynolds-averaged NavierStokes Models . . . . . . . . . . . . . . . . . . . . 82 2.2 The Kolmogorov Theory of Turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . 102 2.2.1 Kolmogorovs universality assumptions . . . . . . . . . . . . . . . . . . . . 102 2.2.2 Hypotheses employed by Frisch [79] . . . . . . . . . . . . . . . . . . . . . . . 103 2.2.3 Principal results of the K41 theory . . . . . . . . . . . . . . . . . . . . . . . . 105 2.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 3 Large-Eddy Simulation and Multi-Scale Methods 113 3.1 Large-Eddy Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 3.1.1 Comparison of DNS, LES and RANS methods . . . . . . . . . . . . . . . . . 114 3.1.2 The LES decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 3.1.3 Derivation of the LES filtered equations . . . . . . . . . . . . . . . . . . . . . 117 3.1.4 Subgrid-scale models for LES . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 3.1.5 Summary of basic LES methods . . . . . . . . . . . . . . . . . . . . . . . . . 127 3.2 Dynamical Systems and Multi-Scale Methods . . . . . . . . . . . . . . . . . . . . . . 127 3.2.1 Some basic tools from the theory of dynamical systems . . . . . . . . . . . . 127....
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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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lctr-notes634 - INTRODUCTORY LECTURES on TURBULENCE...

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