In this monograph, Verma systematically describes various techniques of energy transfers in turbulence. These tools include mode-to-mode transfers, uxes, shell-to-shell and ring-to-ring transfers of energy, as well as enstrophy, kinetic helicity, and magnetic helicity. After developing the framework, the author employs them to turbulence in hydrodynamics, magnetohydrodynamics, passive scalar, buoyancy-driven ows, rotating ows, active scalar and vector, compressible ows, etc. The book describes energy transfers in both real and Fourier space, but the focus is on the latter. The energy transfer diagnostics provide many valuable insights, which have been described throughout the book.
Additional Info
  • Publisher: Cambridge University Press
  • Language: English
  • ISBN : 9781107176195
  • Chapter 1

    Energy Transfers in Fluid Flows

    Turbulence remains an unsolved problem due to the complex nonlinear interactions among a large number of multiscale structures. For hydrodynamic turbulence, Kolmogorov's theory provides quantitative measures of energy contents of the uid structures and energy ux. However, this theory based on real space description does not quantify various scale-by-scale energy transfers. In addition, generalisations of Kolmogorov's theory to more complex system|magnetohydrodynamic and buoyancy-driven turbulence, anisotropic ows, etc.|are quite involved. Fortunately, spectral or Fourier space description, which is the theme of this monograph, overcomes some of these deciencies.

  • Chapter 2

    Preface

    Preface

  • Chapter 3

    Acknowledgements

    Acknowledgements

  • Chapter 4

    Chapter 1 - Introduction Price 0.14  |  0.14 Rewards Points

    Introduction :

    1.1 A Generic Nonlinear Equation

    1.2 Outline of the Book

  • Chapter 5

    Chapter 2 - Basics of Hydrodynamics Price 0.14  |  0.14 Rewards Points

    • 2.1 Governing Equations of Incompressible Flows
    • 2.2 Vorticity and its Equation
    • 2.3 Quadratic Quantities in Hydrodynamics
    • 2.4 Conservation Laws in Hydrodynamics
    • Further Reading
    • Exercises
  • Chapter 6

    Chapter 3 - Fourier Space Description of Hydrodynamics Price 0.14  |  0.14 Rewards Points

    • 3.1 Fourier Transform and its Properties
    • 3.2 Flow Equations in Fourier Space
    • 3.3 Vorticity, Kinetic Helicity, and Enstrophy
    • Further Reading
    • Exercises
  • Chapter 7

    Chapter 4 - Energy Transfers in Hydrodynamic Flows Price 0.14  |  0.14 Rewards Points

    • 4.1 Mode-to-mode Energy Transfers in Hydrodynamics
    • 4.1.1 A physical argument
    • 4.1.2 A mathematical argument based on tensor analysis
    • 4.2 Energy Transfers in the Presence of Many Triads
    • 4.3 Energy Transfers and Equations of Motion for a Two-dimensional Flow
    • 4.4 Spectral Energy Flux
    • 4.5 Variable Energy Flux
    • 4.6 Equivalence between Various Formulas of Energy Flux
    • 4.7 Shell-to-shell Energy Transfers
    • 4.8 Turbulent Energy Flux and Arrow of Time
    • 4.9 Spectral Decomposition, Energy Transfers, and Amplitude Equations
    • 4.10 Numerical Simulations Using Spectral Method
    • 4.11 Computation of Energy Transfers Using Data
    • Further Reading
    • Exercises
  • Chapter 8

    Chapter 5 - Energy Spectrum and Flux of 3D Hydrodynamics Price 0.14  |  0.14 Rewards Points

    • 5.1 Kolmogorov's Theory for 3D Hydrodynamic Turbulence in Spectral Space
    • 5.2 Insights from Kolmogorov's Theory of Turbulence
    • 5.3 Numerical Verication of Kolmogorov's Theory
    • 5.4 Limitations of Kolmogorov's Theory of Turbulence
    • 5.5 Energy Spectrum of Turbulent Flow in the Dissipative Regime
      5.5.1 Pao's model for the inertial{dissipation range of turbulence
      5.5.2 Pope's model for the inertial{dissipation range of turbulence
    • 5.6 Energy Spectrum and Flux for Laminar Flows
    • 5.7 Heisenberg's Theory of Turbulence
    • Further Reading
    • Exercises
  • Chapter 9

    Chapter 6 - Enstrophy Transfers in Hydrodynamics Price 0.14  |  0.14 Rewards Points

    • 6.1 Mode-to-mode Enstrophy Transfers in Hydrodynamics
      6.1.1 Derivation of mode-to-mode enstrophy transfer S!!(k0jpjq)
      6.1.2 Derivation of mode-to-mode enstrophy transfer S!u(k0jpjq)
    • 6.2 Mode-to-mode Enstrophy Transfers in 2D Hydrodynamics
    • 6.3 Enstrophy Transfers for Many Triads
    • 6.4 Enstrophy Fluxes
    • 6.5 Shell-to-shell Enstrophy Transfer
    • 6.6 Numerical Results on Enstrophy Fluxes
    • Further Reading
    • Exercises
  • Chapter 10

    Chapter 7 - Two-dimensional Turbulence Price 0.14  |  0.14 Rewards Points

    • 7.1 Conservation Laws; Energy and Enstrophy Transfers in 2D Hydrodynamics
    • 7.2 Kraichnan's Theory for 2D Hydrodynamic Turbulence
    • 7.3 Subtleties in Energy and Enstrophy Fluxes
    • 7.4 Verication of 2D Hydrodynamic Turbulence Models Using Numerical Simulations
    • Further Reading
    • Exercises
  • Chapter 11

    Chapter 8 - Helical Turbulence Price 0.14  |  0.14 Rewards Points

    • 8.1 Mode-to-mode Kinetic Helicity Transfers in Hydrodynamics
    • 8.2 Flux and Shell-to-shell Transfers of Kinetic Helicity
    • 8.3 Phenomenology of Helical Turbulence
    • 8.4 Numerical Verication of Kinetic Helicity Spectrum and Flux
    • Further Reading
  • Chapter 12

    Chapter 9 - Craya- Herring and Helical Basis Price 0.14  |  0.14 Rewards Points

    • 9.1 Craya-Herring Basis for Hydrodynamics
    • 9.2 Equations of Motion in Craya-Herring Basis
    • 9.3 Energy Transfer Functions in Craya-Herring Basis
    • 9.4 Fluxes in Craya-Herring Basis
    • 9.5 Helical Decomposition
    • 9.6 Helical Modes
    • 9.7 Equations of Motion in Helical Basis
    • 9.8 Mode-to-mode Transfer Functions in Helical Basis
    • 9.9 Fluxes and Shell-to-shell Energy Transfers in Helical Basis
    • Further Reading
    • Exercises
  • Chapter 13

    Chapter 10 - Field-theoretic Treatment of Energy Transfers Price 0.14  |  0.14 Rewards Points

    • 10.1 Correlation Functions in Homogeneous and Isotropic Turbulence
    • 10.2 Field-theoretic Treatment of Mode-to-mode Kinetic Energy Transfers and Flux
    • 10.3 Energy and Enstrophy Transfers in 2D Hydrodynamic Turbulence
    • 10.4 Kinetic Energy and Helicity Transfers in Helical Turbulence
    • Further Reading
    • Exercises
  • Chapter 14

    Chapter 11 - Energy Transfers in Anisotropic Flows Price 0.14  |  0.14 Rewards Points

    • 11.1 Ring Spectrum for Spherical Rings
    • 11.2 Ring Spectrum for Cylindrical Rings
    • 11.3 Ring-to-ring Energy Transfers
    • 11.4 Anisotropic Energy Fluxes, and uk $ u? Energy Exchange
    • Further Reading
  • Chapter 15

    Chapter 12 - Turbulence Properties in Real Space and K41 Theory Price 0.14  |  0.14 Rewards Points

    • 12.1 Second Order Correlation Functions
    • 12.2 Third Order Correlation and Structure Functions
    • 12.3 Kolmogorov's Theory of Turbulence: Four-fth Law
    • 12.4 Another Derivation of Four-fth Law|Frisch (1995)
    • 12.5 Comparison with Spectral Theory
    • 12.6 Higher Order Structure Functions of Hydrodynamic Turbulence
    • Further Reading
  • Chapter 16

    Chapter 13 - Energy Transfers in Flows with Scalars Price 0.14  |  0.14 Rewards Points

    • 13.1 Governing Equations
    • 13.2 Mode-to-mode Scalar Energy Transfers
    • 13.3 Flux and Shell-to-shell Transfers for Scalar Turbulence
    • 13.4 Variable Scalar Energy Flux
    • 13.5 Scalar Field in Craya{Herring Basis
    • Exercises
  • Chapter 17

    Chapter 14 - Flows with a Passive Scalar Price 0.14  |  0.14 Rewards Points

    • 14.1 Governing Equations
    • 14.2 Phenomenology of Passive Scalar Turbulence
    • 14.3 Various Regimes of a Passive Scalar Flow
    • 14.4 Numerical Simulations of Passive Scalar Turbulence
    • 14.5 Third Order Structure Function for Passive Scalar Turbulence:
      Four-third Law
    • 14.6 Field-theoretic Treatment of Passive Scalar Turbulence
    • Further Reading
    • Exercises
  • Chapter 18

    Chapter 15 - Stably Stratied Turbulence Price 0.14  |  0.14 Rewards Points

    • 15.1 Governing Equations in Real Space
    • 15.2 Governing Equations in Fourier Space
    • 15.3 Energy Transfers and Fluxes for Stably Stratied Turbulence
    • 15.4 Various Regimes of Stably Stratied Turbulence
    • 15.5 Stably Stratied Turbulence with Moderate Buoyancy
    • 15.6 Stably Stratied Turbulence with Strong Buoyancy
    • Further Reading
  • Chapter 19

    Chapter 16 - Thermal Convection Price 0.14  |  0.14 Rewards Points

    • 16.1 Governing Equations
    • 16.2 Governing Equations in Fourier Space, Energy Transfers, and Fluxes
    • 16.3 Structure of Temperature Field in Thermal Convection
    • 16.4 Phenomenology of Turbulent Thermal Convection
    • 16.5 Structure Functions of Turbulent Thermal Convection
    • 16.6 Numerical Verication of the Phenomenology of Turbulent Thermal Convection
    • 16.7 Forcing, Energy Dissipation, and Drag Reduction in Turbulent Convection
    • 16.8 Anisotropy in Turbulent Thermal Convection
    • 16.9 Various Regimes of Thermal Convection
    • 16.10 Two-dimensional Turbulent Thermal Convection
    • Further Reading
  • Chapter 20

    Chapter 17 - A More Complex Example of an Active Scalar Binary Fluid Mixture Price 0.14  |  0.14 Rewards Points

    • 17.1 Dynamics of a Binary Fluid Mixture
  • Chapter 21

    Chapter 18 - Energy Transfers in Flows with Vectors Price 0.14  |  0.14 Rewards Points

    • 18.1 Governing Equations
    • 18.2 Mode-to-mode Vector Energy Transfers and Energy Fluxes
    • 18.3 Variable Vector Energy Flux
    • 18.4 Vector Flow in Craya-Herring Basis
    • 18.5 Energy Transfers in Craya-Herring and Helical Basis
  • Chapter 22

    Chapter 19 - Flow with a Passive Vector Price 0.14  |  0.14 Rewards Points

    • 19.1 Governing Equations
    • 19.2 Phenomenology of a Passive Vector Turbulence
    • 19.3 Various Regimes of a Passive Vector Flow
  • Chapter 23

    Chapter 20 - Magnetohydrodynamics Formalism Price 0.14  |  0.14 Rewards Points

    • 20.1 Governing Equations in Real Space
    • 20.2 Conservation Laws
    • 20.3 Governing Equations in Fourier Space
    • 20.4 Alfven Waves
    • 20.5 MHD Equations in Craya-Herring Basis
    • 20.6 MHD Equations in Helical Basis
    • 20.7 Nondimensionalized MHD Equations
    • Further Reading
    • Exercises
  • Chapter 24

    Chapter 21 - Energy Transfers in MHD Price 0.14  |  0.14 Rewards Points

    • 21.1 Combined Energy Transfers in MHD
    • 21.2 Mode-to-mode Energy Transfers in MHD
    • 21.3 Mode-to-mode Transfers for Elsaser Variables
    • 21.4 Miscellaneous Transfers in MHD
    • 21.5 Transfers for Many Triads and Fluxes
    • 21.6 Variable Energy Fluxes and Conserved Fluxes of MHD Turbulence
    • 21.7 Shell-to-shell Transfers in MHD
    • 21.8 Energy Transfers in Craya-Herring Basis
    • 21.9 Energy Transfers in Helical Basis
    • Further Reading
    • Exercises
  • Chapter 25

    Chapter 22 - Models of MHD Turbulence Price 0.14  |  0.14 Rewards Points

    • 22.1 Models of MHD Turbulence
    • 22.2 Third Order Structure Function: Four-third Law
    • 22.3 Higher Order Structure Functions of MHD Turbulence
    • 22.4 Scaling of Cross Helicity and Magnetic Helicity
    • 22.5 MHD Turbulence for Small and Large Prandtl Numbers
    • 22.6 Validation Using Solar Wind
    • 22.7 Validation Using Numerical Simulations
    • 22.8 MHD Turbulence in the Presence of a Mean Magnetic Field
    • Further Reading
  • Chapter 26

    Chapter 23 - Dynamo Magnetic Field Generation in MHD Price 0.14  |  0.14 Rewards Points

    • 23.1 Denitions
    • 23.2 Anti-dynamo Theorems
    • 23.3 Energetics of a Dynamo
    • 23.4 Kinematic Dynamos
    • 23.5 Dynamic Dynamos
    • 23.6 Dynamo Transition and Bifurcation Analysis
    • 23.7 Energy Transfers in Turbulent Dynamos
    • 23.8 Role of Helicities in Dynamos
    • 23.9 Analogy between the Vorticity and Magnetic Fields
    • 23.10 Turbulent Drag Reduction in MHD
    • Further Reading
    • Exercises
  • Chapter 27

    Chapter 24 - Phenomenology of Quasi-Static MHD Turbulence Price 0.14  |  0.14 Rewards Points

    • 24.1 Governing Equations
    • 24.2 Distribution and Spectrum of Kinetic Energy
    • 24.3 Energy Transfers in Quasi-Static MHD
    • Further Reading
  • Chapter 28

    Chapter 25 - Electron Magnetohydrodynamics Price 0.14  |  0.14 Rewards Points

    • 25.1 Governing Equations
    • 25.2 Fourier Space Description
    • 25.3 Phenomenology of EMHD Turbulence
    • 25.3.1 kde 1
    • 25.3.2 kde 1
    • 25.4 Simplied Version
    • Further Reading
  • Chapter 29

    Chapter 26 - Rotating Turbulence Price 0.14  |  0.14 Rewards Points

    • 26.1 Governing Equations
    • 26.2 Properties of Linear Rotating Hydrodynamics
    • 26.3 Nonlinear Regime in Rotating Flows
    • 26.4 Phenomenology of Rotating Turbulence
    • 26.5 Experimental and Numerical Results on Rotating Turbulence
    • Further Reading
  • Chapter 30

    Chapter 27 - Flow with a Tensor Price 0.14  |  0.14 Rewards Points

    • 27.1 Governing Equations
    • 27.2 Mode-to-mode Tensor Energy Transfer and Tensor Energy Flux
    • 27.3 Energy Spectrum and Flux in a Passive Tensor
    • 27.4 Flow with an Active Tensor Field: FENE-p Model
    • 27.5 Turbulent Drag Reduction in Polymeric Flows
    • Further Reading
  • Chapter 31

    Chapter 28 - Shell Models of Turbulence Price 0.14  |  0.14 Rewards Points

    • 28.1 Shell Model for Hydrodynamic Turbulence
    • 28.2 Shell Model for Scalar, Vector, and Tensor Flows
    • Further Reading
  • Chapter 32

    Chapter 29 - Burgers Turbulence Price 0.14  |  0.14 Rewards Points

    • 29.1 Governing Equations
    • 29.2 Energy Transfers in Burgers Turbulence
    • 29.3 Phenomenology of Burgers Turbulence
    • Further Reading
  • Chapter 33

    Chapter 30 - Compressible Turbulence Price 0.14  |  0.14 Rewards Points

    • 30.1 Governing Equations
    • 30.2 Linear Compressible Flow; Sound Waves
    • 30.3 Nearly Incompressible Flow
    • 30.4 Fully Compressible Turbulence: Burgers Turbulence Revisited
    • 30.5 Equation of Motion of a Compressible Flow in Craya-Herring Basis
    • 30.6 Energy Transfers in Compressible Flows
    • Further Reading
  • Chapter 34

    Chapter 31 - Miscellaneous Applications of Energy Transfers Price 0.14  |  0.14 Rewards Points

    • 31.1 Variable Enstrophy Flux in 2D Turbulence with Ekman Friction
    • 31.2 Energy Transfers in Gyrokinetic Plasma Turbulence
    • 31.3 Energy Transfers in Spherical Geometry
    • Further Reading
  • Chapter 35

    Chapter 32 - Conclusions Price 0.14  |  0.14 Rewards Points

    Conclusions

  • Chapter 36

    Appendix

    • Appendix A Power Law Physics
    • Further Reading
    • Appendix B Wealth Distribution and Cascade in an Economy
    • Further Reading
    • Appendix C Renormalization Group Analysis of Hydrodynamic Turbulence
    • Further Reading
  • Chapter 37

    Notation

    Notation

  • Chapter 38

    References

    References

  • Chapter 39

    Subject Index

    Subject Index

About the Author

Mahendra K. Verma is a leading researcher in the eld of turbulence. Presently he is a Professor at the Physics Department of Indian Institute of Technology Kanpur, India. He is a recipient of Swarnajayanti fellowship, INSA Teachers Award, and Dr APJ Abdul Kalam Cray HPC Award. In addition to this book, he has authored Introduction to Mechanics and Physics of Buoyant Flows: From Instabilities to Turbulence. His other research interests include nonlinear dynamics, high-performance computing, and non-equilibrium statistical physics. view complete profile