Origin of Turbulence: Energy Gradient Theory

Preface
Contents
About the Author
Abbreviations
Nomenclature
Greek Symbols
Part I Fundamentals
1 Introduction
1.1 Advances of Research on Turbulence
1.2 Theories of Flow Stability
1.2.1 Linear Stability Theory
1.2.2 Energy Method (Reynolds-Orr)
1.2.3 Nonlinear Stability Theory
1.2.4 Weakly Nonlinear Stability Theory
1.2.5 Secondary Instability Theory
1.2.6 Nonmodal Stability Theory
1.2.7 Energy Gradient Theory
1.3 Definition of Main Terminologies
1.4 Classification of Turbulence
References
2 Equations of Fluid Flow
2.1 Equation of Continuity
2.2 Equation of Momentum
2.3 Equation of Energy
2.4 Equation of Total Mechanical Energy
2.5 Work Done by Shear Stress
References
3 Fundamental of Stability of Parallel Flows
3.1 Stability of Inviscid Parallel Flow
3.1.1 Introduction
3.1.2 Rayleigh Equation
3.1.3 Rayleigh Theorem
3.1.4 Fjϕrtoft Theorem
3.2 Stability of Viscous Parallel Flow
3.2.1 Orr-Sommerfeld Equation
3.2.2 Critical Reynolds Number
3.2.3 Role of Viscosity in Linear Instability
3.3 Energy Method (Reynolds-Orr) and Its Applications
3.4 Scenario of Transition from Laminar Flow to Turbulence
3.4.1 Receptivity
3.4.2 Secondary Instability
3.4.3 Transient Growth
3.4.4 Travelling Waves
3.4.5 Bypass Transition and Breakdown
References
Part II Essentials of Energy Gradient Theory
4 Energy Gradient Theory for Parallel Flow Stability
4.1 Turbulent Transition as a Classical Physical Problem
4.2 Establishment of the Energy Gradient Theory
4.2.1 Locality of Turbulence Generation
4.2.2 Motivation from Viscoelastic Flow Simulation
4.2.3 Observation of Kinetic Energy Gradient for Flow Instability
4.2.4 Stability Criteria Based on Gradient of Total Mechanical Energy
4.3 Derivation of Energy Gradient Function from Navier–Stokes Equation
4.4 Characteristics and Physical Significance of Energy Gradient Function
4.4.1 Kmax Stands for the Most Dangerous Position
4.4.2 Function K Represents the Direction of the Gradient of TME
4.4.3 Mechanism of Flow Instability at Inflection Point
4.4.4 Role of Viscosity in Flow Stability
4.5 Stability of Poiseuille Flows
4.5.1 Plane Poiseuille Flow
4.5.2 Pipe Poiseuille Flow
4.5.3 Comparison with Experiments for Poiseuille Flows
4.6 Annulus Poiseuille Flow Between Two Coaxial Cylinders
4.6.1 Velocity Distribution and Flow Rate in the Annulus
4.6.2 Calculation of K in the Annulus
4.6.3 Comparison to Experiment at Critical Reynolds Number
4.7 Stability of Plane Couette Flow
4.7.1 Energy Gradient Function of Plane Couette Flow
4.7.2 Instability Mechanism and Disturbance Amplification
4.7.3 Comparison with Experiments at Critical Condition
4.7.4 Flux of Vorticity Rather Than Vorticity to Dominate Transition
4.7.5 Conclusions
4.8 Summary
References
5 Turbulent Transition Through Velocity Discontinuity
5.1 Singularity of Navier–Stokes Equations Leading to Turbulence
5.1.1 Axioms for Fluid Flow and Stability
5.1.2 Singularity of Navier–Stokes Equations in Turbulent Transition
5.1.3 Existence and Smoothness of Solution of Navier–Stokes Equations
5.1.4 New Model of Turbulent Transition
5.1.5 Burst Event in Laminar-Turbulent Flow Transition
5.1.6 Sustenance of Fully Developed Turbulence by Singularities
5.1.7 Questions for Turbulent Transition and Turbulence
5.1.8 Summary
5.2 Solution of Navier–Stokes Equation in Poisson Equation Form
5.2.1 Introduction
5.2.2 Navier–Stokes Equation in Form of Poisson Equation
5.2.3 Solution of N-S Equation for Transitional and Turbulent Flows
5.2.4 Significance of the Singularity in Turbulence
5.2.5 Conclusions
5.3 Linear Stability Theory and Turbulent Transition
5.4 Singular Laminar-Turbulent Transition in Plane Couette Flow
5.4.1 Axiom for Fluid Flow in Shear Driven Flows
5.4.2 Discontinuity in Transitional Flow of Plane Couette Flow
5.4.3 DNS Result Confirming the Critical Condition of Transition
5.4.4 Large-Scale Structure of Turbulence in the Core Region
References
6 Stability and Transition of Boundary Layer Flow
6.1 Overview of Stability Research on Boundary Layer Flow
6.2 Singularity in Boundary Layer Flow on Flat Plate
6.3 Stability of Blasius Boundary Layer Flow
6.4 Simulation Results of Streamwise Velocity Profiles
6.4.1 Overshoot of Velocity at the External Edge of Boundary Layer
6.4.2 Distribution of the Energy Gradient Function
6.5 Turbulent Transition Initiated by Spikes in Velocity Distribution
6.6 Wave/Velocity Interaction in Transitional Flow
6.7 Paths in Natural Transition and Bypass Transition
6.8 Structure of Flow in Transitional and Turbulent Flows
6.8.1 Coherent Structure of Turbulence
6.8.2 Self-Sustenance Mechanism of Turbulence
6.9 Interface of Turbulence/Non-turbulence and Entrainment
6.10 Principle of Energy Transmission in Turbulence
6.10.1 How is the Mechanical Energy Exchanged in Turbulence?
6.10.2 Four Modes of Transfer of Mechanical Energy
6.10.3 “Engines” in Turbulence
References
7 Scaling of Disturbance for Turbulent Transition and Turbulence
7.1 Modeling of Disturbance Evolution in Parallel Shear Flows
7.1.1 Energy Increase Versus Energy Loss in Shear Flow
7.1.2 Energy Exchange in Pulsation Disturbance
7.1.3 Stability Criterion in Shear Flow Under Pulsation Disturbance
7.2 Scaling of Disturbance with Re for Turbulent Transition
7.2.1 Amplitude of Disturbance Versus Reynolds Number
7.2.2 Physical Mechanism of Role of Disturbance in Turbulent Transition
7.3 Criteria for Turbulent Transition in Parallel Flows
7.4 Effect of Disturbance Frequency on Turbulent Transition
7.4.1 Relation of Disturbance Frequency with Amplitude
7.4.2 Comparison with Experimental Data
7.5 Energy Spectrum of Turbulence Including Effect of Reynolds Number
7.5.1 Introduction
7.5.2 Theory of Energy Spectrum in Turbulence
7.5.3 Comparison of Theory with Experiments of Turbulence
7.5.4 Conclusions
7.6 Summary
References
8 Stability of Curved and Swirl Flows
8.1 Energy Gradient Theory for Curved Flows
8.2 Theorems on Flow Stability from Energy Gradient Theory
8.3 Stability of Free Vortex Flow
8.3.1 Mechanical Energy Gradient in Radial Direction
8.3.2 Mechanical Energy Loss Distribution for Free Vortex Flow
8.3.3 Calculation of K for Free Vortex Flow
8.4 Stability of Forced Vortex Flow
8.4.1 Mechanical Energy Gradient in the Radial Direction
8.4.2 Mechanical Energy Loss Distribution for Forced Vortex Flow
8.4.3 Calculation of K for Forced Vortex Flow
8.5 Stability of Swirl Flow with Radial Outflow
8.5.1 Mechanical Energy Gradient in the Radial Direction
8.5.2 Mechanical Energy Loss Distribution for Radial Swirl Flow
8.5.3 Calculation of K for Swirl Flow with Radial Outflow
8.6 Mechanism of Tornado Sustenance
References
9 Stability of Taylor-Couette Flow Between Concentric Rotating Cylinders
9.1 Overview of Researches on Taylor-Couette Flow
9.2 Energy Loss Calculation in Taylor-Couette Flow
9.3 Energy Gradient Theory Applied to Taylor-Couette Flow
9.3.1 Velocity Distribution for Taylor-Couette Flow
9.3.2 Energy Gradient in Streamline–Normal Direction
9.3.3 Energy Loss Distribution for Taylor-Couette Flow
9.3.4 Energy Gradient Function K in Taylor-Couette Flow
9.3.5 Critical Condition for Instability of Laminar Flow
9.3.6 Summary
9.4 Singular Laminar-Turbulent Transition in Taylor-Couette Flow
9.4.1 Principle of Singularity Existence in Taylor-Couette Flow
9.4.2 Transition from Velocity Discontinuity to Random Chaos
9.4.3 Summary
References
10 Methods for Prediction of Turbulent Transition
10.1 Semi-empirical Methods
10.2 Method of Exponent e(n) Based on Linear Stability Solution
10.3 Intermittent Factor Method
10.4 Direct Numerical Simulation (DNS) Method
10.5 Criterion of Singularity for Turbulent Transition
10.6 Energy Gradient Method for Parallel Flows
References
Part III Applications
11 Stability of Flow in Curved Duct
11.1 Introduction
11.2 Stability of Flow in a 90° Bend with Rectangular Cross Section
11.2.1 Bend Model and Mathematical Method
11.2.2 Analysis on the Secondary Flow
11.2.3 Summary
11.3 Stability of Flow in a 180° Bend with Rectangular Cross Section
11.3.1 Geometrical Model and Numerical Method
11.3.2 Calculation of Energy Gradient Function
11.3.3 Effect of Re on Flow Instability
11.3.4 Variation of Instability Along the Streamlines
11.3.5 Summary
11.4 Effects of Aspect Ratio on Hydrodynamic Stability
References
12 Stability of Flow in Wake Behind Circular Cylinder
12.1 Flow Around a Cylinder Between Two Parallel Plates
12.1.1 Introduction
12.1.2 Physical Model and the Numerical Method
12.1.3 Calculation of the Energy Gradient Function
12.1.4 Mechanism of Instability Behind the Cylinder
12.1.5 Summary
12.2 Stability of Flow Around Two Rotating Double Cylinders
12.2.1 Introduction
12.2.2 Physical Model and Numerical Method
12.2.3 Wake Instability Induced by Two Rotating Cylinders
12.2.4 Summary
References
13 Application and Stability of Batchelor Vortex Flow
13.1 Stability of Vortex in Draft Tube of Francis Turbine
13.1.1 Introduction
13.1.2 Draft Tube Geometry and Numerical Method
13.1.3 Velocity Distribution at the Inlet of Draft Tube
13.1.4 Stability Analysis of the Draft Tube
13.1.5 Efficacy Analysis of the Draft Tube
13.1.6 Summary
13.2 Stability of Vortex Flow in a Cyclone Separator
13.2.1 Introduction
13.2.2 Geometric Model and Numerical Method
13.2.3 Analysis on Stability of Flow and Separation Efficiency
13.2.4 Summary
References
14 Buoyancy-Driven Instability and Growth
14.1 Criteria of Buoyancy-Driven Instability and Growth
14.2 Stability of Natural Convection in an Inclined Rectangular Cavity
14.2.1 Introduction
14.2.2 Computational Geometry and Numerical Procedures
14.2.3 Criteria of Instability Based on Rayleigh Number
14.2.4 Validation of Numerical Methods
14.2.5 Instability of Thermal Convection
14.2.6 Summary
14.3 Thermal Natural Convection in a Differentially Heated Cavity
14.3.1 Introduction
14.3.2 Computational Geometry and Numerical Procedures
14.3.3 Validation of the Numerical Algorithm
14.3.4 Physical Mechanism of Flow Instability
14.3.5 Relation of Rayleigh Number to Energy Gradient Magnitude
14.3.6 Summary
14.4 Rayleigh–Taylor Instability and Growth
References
15 Stability of Non-Newtonian Fluid Flows
15.1 Instability of Viscoelastic Flow Past a Cylinder in Channel
15.1.1 Introduction
15.1.2 Governing Equations of Viscoelastic Flows
15.1.3 Convergence of Numerical Method and Drag Coefficient
15.1.4 Inflectional Instability by Normal Stress Gradient
15.1.5 Analysis by Boundary Layer Theory
15.1.6 Conclusion and Summary
15.2 Instability Criterion for Viscoelastic Flow Past a Confined Cylinder
15.2.1 Observations from Simulation Results
15.2.2 Flow Instability Initiation from Energy Gradient Theory
15.2.3 Criterion of Instability in Curved Shear Flows
15.2.4 Critical Deborah Number for the Cylinder Problem
15.2.5 Summary
15.3 Flow Stability of MHD Flow in a Rectangular Duct
15.3.1 Introduction
15.3.2 Equations of System Under Magnet Force
15.3.3 Geometrical Model and Numerical Method
15.3.4 Effect of Magnet Force on Flow Instability
15.3.5 Summary
References
Appendix Axioms, Corollaries, and Theorems
Axioms
Corollaries
Theorems
Index