Energy Transfers in Fluid Flows Multiscale and Spectral Perspectives 1st Edition by Mahendra Verma 1107176190 9781107176195

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ISBN 10: 1107176190

ISBN 13: 9781107176195

Author: Mahendra K. Verma

An up-to-date comprehensive text useful for graduate students and academic researchers in the field of energy transfers in fluid flows. The initial part of the text covers discussion on energy transfer formalism in hydrodynamics and the latter part covers applications including passive scalar, buoyancy driven flows, magnetohydrodynamic (MHD), dynamo, rotating flows and compressible flows. Energy transfers among large-scale modes play a critical role in nonlinear instabilities and pattern formation and is discussed comprehensively in the chapter on buoyancy-driven flows. It derives formulae to compute Kolmogorov’s energy flux, shell-to-shell energy transfers and locality. The book discusses the concept of energy transfer formalism which helps in calculating anisotropic turbulence.

Energy Transfers in Fluid Flows Multiscale and Spectral Perspectives 1st Table of contents:

Part I FORMALISM OF ENERGY TRANSFERS

Chapter 1 Introduction

1.1 A Generic Nonlinear Equation

1.2 Outline of the Book

Chapter 2 Basics of Hydrodynamics

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 3 Fourier Space Description of Hydrodynamics

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 4 Energy Transfers in Hydrodynamic Flows

4.1 ModetomodeEnergy 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 Twodimensional Flow

4.4 Spectral Energy Flux

4.5 Variable Energy Flux

4.6 Equivalence between Various Formulas of Energy Flux

4.7 ShelltoshellEnergy 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 5 Energy Spectrum and Flux of 3D Hydrodynamics

5.1 Kolmogorov’s Theory for 3D Hydrodynamic Turbulence in Spectral Space

5.2 Insights from Kolmogorov’s Theory of Turbulence

5.3 Numerical Verification 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 6 Enstrophy Transfers in Hydrodynamics

6.1 ModetomodeEnstrophy Transfers in Hydrodynamics

6.1.1 Derivation of modetomodeenstrophy transfer Sωω(k’|p|q)

6.1.2 Derivation of modetomodeenstrophy transfer Sωu(k’|p|q)

6.2 ModetomodeEnstrophy Transfers in 2D Hydrodynamics

6.3 Enstrophy Transfers for Many Triads

6.4 Enstrophy Fluxes

6.5 ShelltoshellEnstrophy Transfer

6.6 Numerical Results on Enstrophy Fluxes

Further Reading

Exercises

Chapter 7 Two-dimensional Turbulence

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 Verification of 2D Hydrodynamic Turbulence Models Using Numerical Simulations

Further Reading

Exercises

Chapter 8 Helical Turbulence

8.1 ModetomodeKinetic Helicity Transfers in Hydrodynamics

8.2 Flux and ShelltoshellTransfers of Kinetic Helicity

8.3 Phenomenology of Helical Turbulence

8.4 Numerical Verification of Kinetic Helicity Spectrum and Flux

Further Reading

Chapter 9 Craya―Herring and Helical Basis

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

9.6.1 The helical mode u+

9.6.2 The helical mode u_

9.6.3 Mixture of u+ and u_

9.7 Equations of Motion in Helical Basis

9.8 ModetomodeTransfer Functions in Helical Basis

9.9 Fluxes and Shell-to-shell Energy Transfers in Helical Basis

Further Reading

Exercises

Chapter 10 Field-theoretic Treatment of Energy Transfers

10.1 Correlation Functions in Homogeneous and Isotropic Turbulence

10.2 Field-theoretic Treatment of Mode-to-mode Kinetic Energy Transfers and Flux

10.2.1 Computation of

10.2.2 Computation of

10.2.3 Computation of kinetic energy flux and shell-to-shell kinetic energy transfer

10.2.4 Energy transfers for absolute equilibrium turbulence or Euler turbulence

10.3 Energy and Enstrophy Transfers in 2D Hydrodynamic Turbulence

10.4 Kinetic Energy and Helicity Transfers in Helical Turbulence

Further Reading

Exercises

Chapter 11 Energy Transfers in Anisotropic Flows

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 u‖ ↔u┴ Energy Exchange

Further Reading

Chapter 12 Turbulence Properties in Real Space and K41 Theory

12.1 Second Order Correlation Functions

12.2 Third Order Correlation and Structure Functions

12.3 Kolmogorov’s Theory of Turbulence: Four-fifth Law

12.4 Another Derivation of Four-fifth Law—Frisch (1995)

12.5 Comparison with Spectral Theory

12.6 Higher Order Structure Functions of Hydrodynamic Turbulence

Further Reading

Part II FLOWS WITH SCALARS

Chapter 13 Energy Transfers in Flows with Scalars

13.1 Governing Equations

13.2 ModetomodeScalar Energy Transfers

13.2.1 A physical argument

13.2.2 A mathematical argument

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 14 Flows with a Passive Scalar

14.1 Governing Equations

14.2 Phenomenology of Passive Scalar Turbulence

14.3 Various Regimes of a Passive Scalar Flow

14.3.1 Turbulent regime I: Re >> 1; Pe >> 1; Sc ≤ 1|

14.3.2 Laminar regime: Re < 1; Pe < 1|

14.3.3 Mixed regime I: Re >> 1; Pe < 1|

14.3.4 Mixed regime II: Re > 1|

14.3.5 Turbulent regime II: Re >> 1; Pe >> 1; Sc >> 1|

14.4 Numerical Simulations of Passive Scalar Turbulence

14.4.1 Sc ≈ 1

14.4.2 Sc << 1

14.4.3 Sc >> 1

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 15 Stably Stratified Turbulence

15.1 Governing Equations in Real Space

15.2 Governing Equations in Fourier Space

15.3 Energy Transfers and Fluxes for Stably Stratified Turbulence

15.4 Various Regimes of Stably Stratified Turbulence

15.5 Stably Stratified Turbulence with Moderate Buoyancy

15.5.1 Bolgiano–Obukhov phenomenology

15.5.2 Modified Bolgiano–Obukhov scaling

15.5.3 Numerical results on moderately stratified turbulence

15.6 Stably Stratified Turbulence with Strong Buoyancy

Further Reading

Chapter 16 Thermal Convection

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 Verification of the Phenomenology of Turbulent Thermal Convection

16.6.1 Kinetic energy spectrum and flux; Scalar energy flux

16.6.2 Scalar energy or temperature spectrum

16.6.3 Structure functions

16.6.4 Shell-to-shell energy transfers

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

16.9.1 Re >> 1; Pe >> 1; Pr ≈ 1

16.9.2 Re >> 1; Pr = 0

16.9.3 Re >> 1; Small Pr

16.9.4 Pe >> 1; Pr = ∞

16.10 Two-dimensional Turbulent Thermal Convection

Further Reading

Chapter 17 A More Complex Example of an Active Scalar: Binary Fluid Mixture

17.1 Dynamics of a Binary Fluid Mixture

Part III FLOWS WITH VECTORS

Chapter 18 Energy Transfers in Flows with Vectors

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 19 Flow with a Passive Vector

19.1 Governing Equations

19.2 Phenomenology of a Passive Vector Turbulence

19.3 Various Regimes of a Passive Vector Flow

Chapter 20 Magnetohydrodynamics: Formalism

20.1 Governing Equations in Real Space

20.2 Conservation Laws

20.3 Governing Equations in Fourier Space

20.4 Alfvén 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 21 Energy Transfers in MHD

21.1 Combined Energy Transfers in MHD

21.2 Mode-to-mode Energy Transfers in MHD

21.3 Mode-to-mode Transfers for Elsäser Variables

21.4 Miscellaneous Transfers in MHD

21.4.1 Mode-to-mode magnetic helicity transfers in MHD

21.4.2 Mode-to-mode kinetic helicity transfers in MHD

21.4.3 Mode-to-mode transfers of EA in 2D

21.5 Transfers for Many Triads and Fluxes

21.6 Variable Energy Fluxes and Conserved Fluxes of MHD Turbulence

21.6.1 Kinetic and magnetic energy fluxes

21.6.2 Fluxes for Elsäser fields and magnetic helicity

21.7 ShelltoshellTransfers in MHD

21.8 Energy Transfers in Craya–Herring Basis

21.9 Energy Transfers in Helical Basis

Further Reading

Exercises

Chapter 22 Models of MHD Turbulence

22.1 Models of MHD Turbulence

22.1.1 Kraichnan and Iroshnikov’s model—E(k) ∞ k-3/2

22.1.2 Dobrowonly et al.’s model

22.1.3 Model based on energy fluxes

22.1.4 Goldreich and Sridhar—E(k┴) ~ k┴-5/3

22.1.5 Verma—Effective mean magnetic field and E(k) ∞ k-5/3

22.1.6 Galtier et al.—Weak turbulence and E(k┴) ∞ k┴-2

22.1.7 Boldyrev et al.—Dynamic alignment yields k-3/2 spectrum

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

22.4.1 Scaling of cross helicity

22.4.2 Scaling of magnetic helicity

22.5 MHD Turbulence for Small and Large Prandtl Numbers

22.5.1 Energy spectra of small Pm MHD

22.5.2 Energy spectra of large Pm MHD

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 23 Dynamo: Magnetic Field Generation in MHD

23.1 Definitions

23.2 Anti-dynamo Theorems

23.3 Energetics of a Dynamo

23.4 Kinematic Dynamos

23.4.1 Sixmodemodel—Verma et al. (2008)

23.4.2 Roberts dynamo

23.4.3 A 2D3C helical dynamo model?

23.4.4 A tetrahedron helical dynamo model—Stepanov and Plunian (2018)

23.5 Dynamic Dynamos

23.5.1 Sixmodemodel—Verma et al. (2008) revisited

23.6 Dynamo Transition and Bifurcation Analysis

23.7 Energy Transfers in Turbulent Dynamos

23.7.1 Small Pm dynamos

23.7.2 Large Pm dynamos

23.7.3 Largescaledynamo with forcing at intermediate scale

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 24 Phenomenology of Quasi-static MHD Turbulence

24.1 Governing Equations

24.2 Distribution and Spectrum of Kinetic Energy

24.3 Energy Transfers in QuasiStaticMHD

Further Reading

Chapter 25 Electron Magnetohydrodynamics

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 Simplified Version

25.4.1 Governing equations and conservation laws

25.4.2 Energy transfers in EMHD

Further Reading

Part IV MISCELLANEOUS FLOWS

Chapter 26 Rotating Turbulence

26.1 Governing Equations

26.2 Properties of Linear Rotating Hydrodynamics

26.2.1 Taylor–Proudman theorem

26.2.2 Inertial waves in rotating flows

26.3 Nonlinear Regime in Rotating Flows

26.4 Phenomenology of Rotating Turbulence

26.4.1 Zeman’s phenomenology

26.4.2 Zhou’s phenomenology

26.4.3 Smith and Waleffe’s phenomenology

26.4.4 Kuznetsov–Zakharov–Kolmogorov spectrum

26.4.5 Inferences from the energy transfers in rotating turbulence

26.5 Experimental and Numerical Results on Rotating Turbulence

Further Reading

Chapter 27 Flow with a Tensor

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

27.4.1 Governing equations

27.4.2 Energy spectra and fluxes in the FENE-p model

27.5 Turbulent Drag Reduction in Polymeric Flows

Further Reading

Chapter 28 Shell Models of Turbulence

28.1 Shell Model for Hydrodynamic Turbulence

28.1.1 Shell model

28.1.2 Energy transfers in the shell model

28.2 Shell Model for Scalar, Vector, and Tensor Flows

Further Reading

Chapter 29 Burgers Turbulence

29.1 Governing Equations

29.2 Energy Transfers in Burgers Turbulence

29.3 Phenomenology of Burgers Turbulence

Further Reading

Chapter 30 Compressible Turbulence

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

30.6.1 Equations for modal kinetic and internal energies

30.6.2 Triadic interactions in a compressible flow?

30.6.3 Energy fluxes in compressible turbulence

Further Reading

Chapter 31 Miscellaneous Applications of Energy Transfers

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 32 Conclusions

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

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