Chapter
2.5.2 Poloidal–Toroidal Decomposition and Craya–Herring Frame of Reference
2.5.3 Helical-Mode Decomposition
2.5.4 Use of Projection Operators
2.5.6 Background Nonlinearity in Different Reference Frames
2.6 Anisotropy in Fourier Space
2.6.1 Second-Order Velocity Statistics
2.6.1.1 Directional and Polarization Anisotropy – Intrinsic Form
2.6.1.2 Induced Anisotropic Structure of Arbitrary Second-Order Statistical Quantities
2.6.1.3 Bridging with Dimensionality and Componentality
2.6.2 Some Comments About Higher-Order Statistics
2.7 A Synthetic Scheme of the Closure Problem: Nonlinearity and Nonlocality
3 Incompressible Homogeneous Isotropic Turbulence
3.1 Observations and Measures in Forced and Freely Decaying Turbulence
3.1.1 How to Generate Isotropic Turbulence?
3.1.2 Main Observed Statistical Features of Developed Isotropic Turbulence
3.1.3 Energy Decay Regimes
3.1.4 Coherent Structures in Isotropic Turbulence
3.2 Self-Similar Decay Regimes, Symmetries, and Invariants
3.2.1 Symmetries of Navier–Stokes Equations and Existence of Self-Similar Solutions
3.2.2 Algebraic Decay Exponents Deduced From Symmetry Analysis
3.2.3 Time-Variation Exponent and Inviscid Global Invariants
3.2.4 Refined Analysis Without PLE Hypothesis
3.2.5 Self-Similarity Breakdown
3.2.6 Self-Similar Decay in the Final Region
3.3 Reynolds Stress Tensor and Analysis of Related Equations
3.4 Classical Statistical Analysis: Energy Cascade, Local Isotropy, Usual Characteristic Scales
3.4.1 Double Correlations and Typical Scales
3.4.2 (Very Brief) Reminder About Kolmogorov Legacy, Structure Functions, “Modern” Scaling Approach
3.4.3 Turbulent Kinetic-Energy Cascade in Fourier Space
3.5 Advanced Analysis of Energy Transfers in Fourier Space
3.5.1 The Background Triadic Interaction
3.5.2 Nonlinear Energy Transfers and Triple Correlations
3.5.3 Global and Detailed Conservation Properties
3.5.4 Advanced Analysis of Triadic Transfers and Waleffe’s Instability Assumption
3.5.5 Further Discussions About the Instability Assumption
3.5.6 Principle of Quasi-Normal Closures
3.5.7 EDQNM for Isotropic Turbulence. Final Equations and Results
3.5.7.1 Well-Documented Experimental Data, Moderate Reynolds Number
3.5.7.2 Transfer Term at Increasing Reynolds Number
3.5.7.3 Toward an Infinite Reynolds Number
3.5.7.4 Very Recent Improvements
3.5.7.5 On Instantaneous Energy Transfers
3.5.7.6 Nonlinear Cascade Time Scale, Equilibrium, and Dissipation Asymptotics
3.6 Topological Analysis, Coherent Events, and Related Dynamics
3.6.1 Topological Analysis of Isotropic Turbulence
3.6.2 Vortex Tube: Statistical Properties and Dynamics
3.6.3 Bridging with Turbulence Dynamics and Intermittency
3.7 Nonlinear Dynamics in the Physical Space
3.7.1 On Vortices, Scales, Wavenumbers, and Wave Vectors – What are the Small Scales?
3.7.2 Is There an Energy Cascade in the Physical Space?
3.7.3 Self-Amplification of Velocity Gradients
3.7.4 Non-Gaussianity and Depletion of Nonlinearity
3.8 What are the Proper Features of Three-Dimensional Navier–Stokes Turbulence?
3.8.1 Influence of the Space Dimension: Introduction to d-Dimensional Turbulence
3.8.2 Pure 2D Turbulence and Dual Cascade
3.8.3 Role of Pressure: A View of Burgers’ Turbulence
3.8.4 Sensitivity with Respect to Energy-Pumping Process: Turbulence with Hyperviscosity
4 Incompressible Homogeneous Anisotropic Turbulence: Pure Rotation
4.1 Physical and Numerical Experiments
4.1.1 Brief Review of Experiments, More or Less in the Configuration of Homogeneous Turbulence
4.2.2 Important Nondimensional Numbers. Particular Regimes
4.3 Advanced Analysis of Energy Transfer by DNS
4.4 Balance of RST Equations. A Case Without “Production.” New Tensorial Modeling
4.5 Inertial Waves. Linear Regime
4.5.1 Analysis of Deterministic Solutions
4.5.2 Analysis of Statistical Moments. Phase Mixing and Low-Dimensional Manifolds
4.5.2.1 Single-Time Second-Order Statistics
4.5.2.2 Single-Time Third-Order Statistics
4.6 Nonlinear Theory and Modeling: Wave Turbulence and EDQNM
4.6.1 Full Exact Nonlinear Equations. Wave Turbulence
4.6.2 Second-Order Statistics: Identification of Relevant Spectral-Transfer Terms
4.6.3 Toward a Rational Closure with an EDQNM Model
4.6.4 Recovering the Asymptotic Theory of Inertial Wave Turbulence
4.7 Fundamental Issues: Solved and Open Questions
4.7.1 Eventual Two-Dimensionalization or Not
4.7.2 Meaning of the Slow Manifold
4.7.3 Are Present DNS and LES Useful for Theoretical Prediction?
4.7.4 Is the Pure Linear Theory Relevant?
4.7.5 Provisional Conclusions About Scaling Laws and Quantified Values of Key Descriptors
4.8 Coherent Structures, Description, and Dynamics
5 Incompressible Homogeneous Anisotropic Turbulence: Strain
5.2 Experiments for Turbulence in the Presence of Mean Strain.Kinematics of the Mean Flow
5.2.1 Pure Irrotational Strain, Planar Distortion
5.2.2 Axisymmetric (Irrotational) Strain
5.2.3 The Most General Case for 3D Irrotational Case
5.2.4 More General Distortions. Kinematics of Rotational Mean Flows
5.3 First Approach in Physical Space to Irrotational Mean Flows
5.3.1 Governing Equations, RST Balance, and Single-Point Modeling
5.3.1.2 Axisymmetric Irrotational Strain
5.3.1.3 More General Rotational Strains
5.3.2 General Assessment of RST Single-Point Closures
5.3.3 Linear Response of Turbulence to Irrotational Mean Strain
5.4 The Fundamentals of Homogeneous RDT
5.4.1 Qualitative Trends Induced by the Green’s Function
5.4.2 Results at Very Short Times. Relevance at Large Elapsed Times
5.5 Final RDT Results for Mean Irrotational Strain
5.5.1 General RDT Solution
5.5.2 Linear Response of Turbulence to Axisymmetric Strain
5.6 First Step Toward a Nonlinear Approach
5.7 Nonhomogeneous Flow Cases. Coherent Structures in Strained Homogeneous Turbulence
6 Incompressible Homogeneous Anisotropic Turbulence: Pure Shear
6.1 Physical and Numerical Experiments: Kinetic Energy, RST, Length Scales, Anisotropy
6.1.1 Experimental and Numerical Realizations
6.2 Reynolds Stress Tensor and Analysis of Related Equations
6.3 Rapid Distortion Theory: Equations, Solutions, Algebraic Growth
6.3.1 Some Properties of RDT Solutions
6.3.2 Relevance of Homogeneous RDT
6.4 Evidence and Uncertainties for Nonlinear Evolution: Kinetic-Energy Exponential Growth Using Spectral Theory
6.5 Vortical–Structure Dynamics in Homogeneous Shear Turbulence
6.6 Self-Sustaining Turbulent Cycle in Homogeneous Sheared Turbulence
6.7 Self-Sustaining Processes in Nonhomogeneous Sheared Turbulence: Exact Coherent States and Traveling-Wave Solutions
6.8 Local Isotropy in Homogeneous Shear Flows
7 Incompressible Homogeneous Anisotropic Turbulence: Buoyancy and Stable Stratification
7.1 Observations, Propagating and Nonpropagating Motion. Collapse of Vertical Motion and Layering
7.2 Simplified Equations, Using Navier–Stokes and Boussinesq Approximations, With Uniform Density Gradient
7.2.1 Reynolds Stress Equations With Additional Scalar Variance and Flux
7.2.2 First Look at Gravity Waves
7.3 Eigenmode Decomposition. Physical Interpretation
7.4 The Toroidal Cascade as a Strong Nonlinear Mechanism Explaining the Layering
7.5 The Viewpoint of Modeling and Theory: RDT, Wave Turbulence, EDQNM
7.6 Coherent Structures: Dynamics and Scaling of the Layered Flow, “Pancake” Dynamics, Instabilities
7.6.1 Simplified Scaling Laws
7.6.2 Pancake Structures, Zig-Zag, and Kelvin–Helmholtz Instabilities
8 Coupled Effects: Rotation, Stratification, Strain, and Shear
8.1 Rotating Stratified Turbulence
8.1.1 Basic Triadic Interaction for Quasi-Geostrophic Cascade
8.1.2 About the Case With Small but Nonnegligible f/N Ratio
8.1.3 The QG Model Revisited. Discussion
8.2 Rotation or Stratification With Mean Shear
8.2.1 The Rotating-Shear-Flow Case
8.2.2 The Stratified-Shear-Flow Case
8.2.3 Analogies and Differences Between the Two Cases
8.3 Shear, Rotation, and Stratification. RDT Approach to Baroclinic Instability
8.3.1 Physical Context, the Mean Flow
8.4 Elliptical Flow Instability From “Homogeneous” RDT
8.5 Axisymmetric Strain With Rotation
8.6 Relevance of RDT and WKB RDT Variants for Analysis of Classical Instabilities
9 Compressible Homogeneous Isotropic Turbulence
9.1 Introduction to Modal Decomposition of Turbulent Fluctuations
9.1.1 Statement of the Problem
9.1.2 Kovasznay’s Linear Decomposition
9.1.3 Weakly Nonlinear Corrected Kovasznay Decomposition
9.1.4 Helmholtz Decomposition and Its Extension
9.1.5 Bridging Between Kovasznay and Helmholtz Decomposition
9.1.6 On the Feasibility of a Fully General Modal Decomposition
9.2 Mean-Flow Equations, Reynolds Stress Tensor, and Energy Balance in Compressible Flows
9.2.2 Simplifications in the Isotropic Case
9.2.3 Quasi-Isentropic Isotropic Turbulence: Physical and Spectral Descriptions
9.3 Different Regimes in Compressible Turbulence
9.3.1 Quasi-Isentropic Turbulent Regime
9.3.1.2 The Relevant Incompressible Limit for Both Spectra of Solenoidal Energy and Pressure Variance
9.3.1.3 Quasi-Inviscid Limit: Toward an Extended Wave-Turbulence Model
9.3.1.4 Introducing Relevant Eddy-Damping. Main Results
9.3.1.5 Additional Discussion About the Modified Decorrelation Function
9.3.1.6 Analytical Fauchet–Bertoglio Model
9.3.1.7 Numerical Experiments
9.3.2 Weakly Compressible Thermal Regime
9.3.2.1 Asymptotic Analysis and Possible Thermal Regimes
9.3.2.2 Statistical Equilibrium States
9.3.2.3 Numerical Observations
9.3.3 Nonlinear Subsonic Regime
9.3.3.1 Conditions for Occurrence of Shocklets
9.3.3.2 Energy Budget and Shocklet Influence
9.3.3.3 Enstrophy Budget and Shocklet Influence
9.3.3.4 Statistical Equilibrium State
9.4 Structures in the Physical Space
9.4.1 Turbulent Structures in Compressible Turbulence
9.4.2 A Probabilistic Model for Shocklets
10 Compressible Homogeneous Anisotropic
10.1 Effects of Compressibility in Free-Shear Flows. Observations
10.1.1 RST Equations and Single-Point Modeling
10.1.2 Preliminary Linear Approach: Pressure-Released Limit and Irrotational Strain
10.2 A General Quasi-Isentropic Approach to Homogeneous Compressible Shear Flows
10.2.1 Governing Equations and Admissible Mean Flows
10.2.2 Properties of Admissible Mean Flows
10.2.3 Linear Response in Fourier Space. Governing Equations
10.2.3.1 Recovering the Acoustic Regime
10.2.3.2 Recovering the Solenoidal Limit
10.2.3.3 Irrotational Mean-Strain Case
10.3 Incompressible Turbulence With Compressible Mean-Flow Effects: Compressed Turbulence
10.4 Compressible Turbulence in the Presence of Pure Plane Shear
10.4.1 Qualitative Results
10.4.2 Discussion of Results
10.4.3 Toward a Complete Linear Solution
10.5 Perspectives and Open Issues
10.5.1 Homogeneous Shear Flows
10.5.2 Perspectives Toward Inhomogeneous Shear Flows
10.6 Topological Analysis, Coherent Events and Related Dynamics
10.6.1 Nonlinear Dynamics in the Subsonic Regime
10.6.2 Topological Analysis of the Rate-of-Strain Tensor
10.6.3 Vortices, Shocklets, and Dynamics
11 Isotropic Turbulence–Shock Interaction
11.1 Brief Survey of Existing Interaction Regimes
11.1.1 Destructive Interactions
11.1.2 Nondestructive Interactions
11.2 Linear Nondestructive Interaction
11.2.1 Shock Modeling and Jump Relations
11.2.2 Introduction to the Linear Interaction Approximation Theory
11.2.3 Vortical Turbulence–Shock Interaction
11.2.4 Acoustic Turbulence–Shock Interaction
11.2.5 Mixed Turbulence–Shock Interaction
11.2.5.1 Influence of the Upstream Entropy Fluctuations
11.2.5.2 Influence of the Upstream Acoustic Fluctuations
11.2.6 On the Use of RDT for Linear Nondestructive Interaction Modeling
11.3 Nonlinear Nondestructive Interactions
11.3.1 Turbulent Jump Conditions for the Mean Field
11.3.2 Jump Conditions for an Incident Isotropic Turbulence
12 Linear Interaction Approximation for Shock–Perturbation Interaction
12.1 Shock Description and Emitted Fluctuating Field
12.2 Calculation of Wave Vectors of Emitted Waves
12.2.2 Incident Entropy and Vorticity Waves
12.2.2.1 Emitted Entropy and Vorticity Waves
12.2.2.2 Emitted Acoustic Waves—Propagative and Nonpropagative Regimes
12.2.3 Incident Acoustic Waves
12.2.3.1 Fast and Slow Waves
12.2.3.2 Emitted Entropy and Vorticity Waves
12.2.3.3 Emitted Acoustic Waves
12.3 Calculation of Amplitude of Emitted Waves
12.3.1 General Decompositions of the Perturbation Field
12.3.2 Calculation of Amplitudes of Emitted Waves
12.4 Reconstruction of the Second-Order Moments
12.4.1 Case of a Single Incident Wave
12.4.2 Case of an Incident Turbulent Isotropic Field
12.5 A posteriori Assessment of LIA
13 Linear Theories. From Rapid Distortion Theory to WKB Variants
13.1 Rapid Distortion Theory for Homogeneous Turbulence
13.1.1 Solutions for ODEs in Orthonormal Fixed Frames of Reference
13.1.2 Using Solenoidal Modes for a Green’s Function with a Minimal Number of Components
13.1.3 Prediction of Statistical Quantities
13.1.3.1 Initial-Value Problem or Forcing?
13.1.4 RDT for Two-Time Correlations
13.2 Zonal RDT and Short-Wave Stability Analysis
13.2.1 Irrotational Mean Flows
13.2.2 Zonal Stability Analysis With Disturbances Localized Around Base-Flow Trajectories
13.2.3 Using Characteristic Rays Related to Waves Instead of Trajectories
13.3 Application to Statistical Modeling of Inhomogeneous Turbulence
13.3.1 Transport Models Along Mean Trajectories
13.3.2 Semiempirical Transport “Shell” Models
13.4 Conclusions, Recent Perspectives Including Subgrid-Scale Dynamics Modeling
14 Anisotropic Nonlinear Triadic Closures
14.1 Canonical HIT, Dependence on the Eddy Damping for the Scaling of the Energy Spectrum in the Inertial Range
14.2 Solving the Linear Operator to Account for Strong Anisotropy
14.2.1 Random and Averaged Nonlinear Green’s Functions
14.2.2 Homogeneous Anisotropic Turbulence with a Mean Flow
14.3 A General EDQN Closure. Different Levels of Markovianization
14.3.2 A Simplified Version: EDQNM1
14.3.2.1 Recovering the Conventional 2D Case With Additional Jetal Mode
14.3.3 The Most Sophisticated Version: EDQNM3
14.4 Application of Three Versions to the Rotating Turbulence
14.5 Other Cases of Flows With and Without Production
14.5.1 Effects of the Distorting Mean Flow
14.5.1.1 Hyperbolic and Elliptic Cases
14.5.2 Flows Without Production Combining Strong and Weak Turbulence
14.5.2.1 Buoyant Flows in a Stably Stratified Fluid
14.5.2.2 Weakly Compressible Isotropic Turbulence
14.5.3 Role of the Nonlinear Decorrelation Time Scale
14.6 Connection with Self-Consistent Theories: Single Time or Two Time?
14.7 Applications to Weak Anisotropy
14.7.1 A Self-Consistent Representation of the Spectral Tensor for Weak Anisotropy
14.7.2 Brief Discussion of Concepts, Results, and Open Issues
14.8 Open Numerical Problems
15 Conclusions and Perspectives
15.1 Homogenization of Turbulence. Local or Global Homogeneity? Physical Space or Fourier Space?
15.2 Linear Theory, “Homogeneous” RDT, WKB Variants, and LIA
15.3 Multipoint Closures for Weak and Strong Turbulence
15.3.1 The Wave-Turbulence Limit
15.3.2 Coexistence of Weak and Strong Turbulence, With Interactions
15.3.3 Revisiting Basic Assumptions in MPC
15.4 Structure Formation, Structuring Effects, and Individual Coherent Structures
15.5 Anisotropy Including Dimensionality, a Main Theme
15.6 Deriving Practical Models
Homogeneous Turbulence Dynamics
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