TURBULENCE, ENTROPY AND DYNAMICS Lecture Notes, UPC 2014 Jose M. Redondo Contents 1 Turbulence 1 1.1 Features ................................................ 2 1.2 Examples of turbulence ........................................ 3 1.3 Heat and momentum transfer ..................................... 4 1.4 Kolmogorov’s theory of 1941 ..................................... 4 1.5 See also ................................................ 6 1.6 References and notes ......................................... 6 1.7 Further reading ............................................ 7 1.7.1 General ............................................ 7 1.7.2 Original scientific research papers and classic monographs .................. 7 1.8 External links ............................................. 7 2 Turbulence modeling 8 2.1 Closure problem ............................................ 8 2.2 Eddy viscosity ............................................. 8 2.3 Prandtl’s mixing-length concept .................................... 8 2.4 Smagorinsky model for the sub-grid scale eddy viscosity ....................... 8 2.5 Spalart–Allmaras, k–ε and k–ω models ................................ 9 2.6 Common models ........................................... 9 2.7 References ............................................... 9 2.7.1 Notes ............................................. 9 2.7.2 Other ............................................. 9 3 Reynolds stress equation model 10 3.1 Production term ............................................ 10 3.2 Pressure-strain interactions ...................................... 10 3.3 Dissipation term ........................................... 10 3.4 Diffusion term ............................................ 10 3.5 Pressure-strain correlation term .................................... 10 3.6 Rotational term ............................................ 11 3.7 Advantages of RSM .......................................... 11 3.8 Disadvantages of RSM ........................................ 11 3.9 See also ................................................ 11 i ii CONTENTS 3.10 See also ................................................ 11 3.11 References ............................................... 11 3.12 Bibliography .............................................. 11 4 Boundary layer 12 4.1 Aerodynamics ............................................. 12 4.2 Naval architecture ........................................... 13 4.3 Boundary layer equations ....................................... 13 4.4 Turbulent boundary layers ....................................... 14 4.5 Heat and mass transfer ........................................ 14 4.6 Convective transfer constants from boundary layer analysis ...................... 15 4.7 Boundary layer turbine ........................................ 16 4.8 See also ................................................ 16 4.9 References ............................................... 17 4.10 External links ............................................. 17 5 Similitude (model) 18 5.1 Overview ............................................... 18 5.2 An example .............................................. 19 5.3 Typical applications .......................................... 19 5.4 Notes ................................................. 20 5.5 See also ................................................ 20 5.6 References .............................................. 20 5.7 External links ............................................. 20 6 Lagrangian and Eulerian specification of the flow field 21 6.1 Description .............................................. 21 6.2 Substantial derivative ......................................... 21 6.3 See also ................................................ 22 6.4 Notes ................................................. 22 6.5 References ............................................... 22 7 Lagrangian mechanics 23 7.1 Conceptual framework ........................................ 23 7.1.1 Generalized coordinates .................................... 23 7.1.2 D'Alembert’s principle and generalized forces ........................ 24 7.1.3 Kinetic energy relations .................................... 24 7.1.4 Lagrangian and action ..................................... 25 7.1.5 Hamilton’s principle of stationary action ........................... 25 7.2 Lagrange equations of the first kind .................................. 26 7.3 Lagrange equations of the second kind ................................ 26 7.3.1 Euler–Lagrange equations ................................... 26 7.3.2 Derivation of Lagrange’s equations .............................. 26 CONTENTS iii 7.3.3 Dissipation function ...................................... 27 7.3.4 Examples ........................................... 27 7.4 Extensions of Lagrangian mechanics ................................. 29 7.5 See also ................................................ 30 7.6 References ............................................... 30 7.7 Further reading ............................................ 30 7.8 External links ............................................. 31 8 Hamiltonian mechanics 32 8.1 Overview ............................................... 32 8.1.1 Basic physical interpretation ................................. 32 8.1.2 Calculating a Hamiltonian from a Lagrangian ........................ 32 8.2 Deriving Hamilton’s equations ..................................... 33 8.3 As a reformulation of Lagrangian mechanics ............................. 33 8.4 Geometry of Hamiltonian systems .................................. 34 8.5 Generalization to quantum mechanics through Poisson bracket .................... 34 8.6 Mathematical formalism ........................................ 35 8.7 Riemannian manifolds ......................................... 35 8.8 Sub-Riemannian manifolds ...................................... 36 8.9 Poisson algebras ............................................ 36 8.10 Charged particle in an electromagnetic field .............................. 36 8.11 Relativistic charged particle in an electromagnetic field ........................ 36 8.12 See also ................................................ 37 8.13 References ............................................... 37 8.13.1 Footnotes ........................................... 37 8.13.2 Sources ............................................ 37 8.14 External links ............................................. 37 9 Classical mechanics 38 9.1 History ................................................. 39 9.2 Description of the theory ....................................... 40 9.2.1 Position and its derivatives .................................. 41 9.2.2 Forces; Newton’s second law ................................. 42 9.2.3 Work and energy ....................................... 43 9.2.4 Beyond Newton’s laws .................................... 43 9.3 Limits of validity ........................................... 43 9.3.1 The Newtonian approximation to special relativity ...................... 44 9.3.2 The classical approximation to quantum mechanics ..................... 44 9.4 Branches ................................................ 44 9.5 See also ................................................ 45 9.6 Notes ................................................. 45 9.7 References ............................................... 45 iv CONTENTS 9.8 Further reading ............................................ 45 9.9 External links ............................................. 46 10 Entropy (information theory) 47 10.1 Introduction .............................................. 47 10.2 Definition ............................................... 48 10.3 Example ................................................ 48 10.4 Rationale ............................................... 49 10.5 Aspects ................................................ 49 10.5.1 Relationship to thermodynamic entropy ........................... 49 10.5.2 Entropy as information content ................................ 50 10.5.3 Data compression ....................................... 50 10.5.4 World’s technological capacity to store and communicate entropic information ....... 51 10.5.5 Limitations of entropy as information content ........................ 51 10.5.6 Limitations of entropy as a measure of unpredictability ................... 51 10.5.7 Data as a Markov process ................................... 52 10.5.8 b-ary entropy ......................................... 52 10.6 Efficiency ............................................... 52 10.7 Characterization ............................................ 52 10.7.1 Continuity ........................................... 52 10.7.2 Symmetry ........................................... 53 10.7.3 Maximum ........................................... 53 10.7.4 Additivity ........................................... 53 10.8 Further properties ........................................... 53 10.9 Extending discrete entropy to the continuous case ........................... 54 10.9.1 Differential entropy ...................................... 54 10.9.2 Relative entropy ........................................ 54 10.10Use in combinatorics ......................................... 55 10.10.1 Loomis-Whitney inequality .................................. 55 10.10.2 Approximation to binomial coefficient ............................ 55 10.11See also ................................................ 55 10.12References ............................................... 56