Homogeneous Turbulence Dynamics

Transcription

1 Homogeneous Turbulence Dynamics PIERRE SAGAUT Universite Pierre et Marie Curie CLAUDE CAMBON Ecole Centrale de Lyon «Hf CAMBRIDGE Щ0 UNIVERSITY PRESS

2 Abbreviations Used in This Book page xvi 1 Introduction Scope of the Book Structure and Contents of the Book 3 Bibliography 9 2 Statistical Analysis of Homogeneous Turbulent Flows: Reminders Background Deterministic Equations Mass Conservation The Navier-Stokes Momentum Equations Incompressible Turbulence First Insight into Compressibility Effects Reminder About Circulation and Vorticity Adding Body Forces or Mean Gradients Briefs About Statistical and Probabilistic Approaches Ensemble Averaging, Statistical Homogeneity Single-Point and Multipoint Moments Statistics for Velocity Increments Application of Reynolds Decomposition to Dynamical Equations Reynolds Stress Tensor and Related Equations RST Equations The Mean Flow Consistent With Homogeneity Homogeneous RST Equations. Briefs About Closure Methods Anisotropy in Physical Space. Single-Point and Two-Point Correlations Spectral Analysis, From Random Fields to Two-Point «Correlations. Local Frame, Helical Modes Second-Order Statistics Poloidal-Toroidal Decomposition and Craya-Herring Frame of Reference Helical-Mode Decomposition Use of Projection Operators Nonlinear Dynamics Background Nonlinearity in Different Reference Frames 36 VII

3 viii Contents 2.6 Anisotropy in Fourier Space Second-Order Velocity Statistics Some Comments About Higher-Order Statistics A Synthetic Scheme of the Closure Problem: Nonlinearity and Nonlocality 43 Bibliography 47 3 Incompressible Homogeneous Isotropic Turbulence Observations and Measures in Forced and Freely Decaying Turbulence How to Generate Isotropic Turbulence? Main Observed Statistical Features of Developed Isotropic Turbulence Energy Decay Regimes Coherent Structures in Isotropic Turbulence Self-Similar Decay Regimes, Symmetries, and Invariants Symmetries of Navier-Stokes Equations and Existence of Self-Similar Solutions Algebraic Decay Exponents Deduced From Symmetry Analysis Time-Variation Exponent and Inviscid Global Invariants Refined Analysis Without PLE Hypothesis Self-Similarity Breakdown Self-Similar Decay in the Final Region Reynolds Stress Tensor and Analysis of Related Equations Classical Statistical Analysis: Energy Cascade, Local Isotropy, Usual Characteristic Scales Double Correlations and Typical Scales (Very Brief) Reminder About Kolmogorov Legacy, Structure Functions, "Modern" Scaling Approach Turbulent Kinetic-Energy Cascade in Fourier Space Advanced Analysis of Energy Transfers in Fourier Space The Background Triadic Interaction Nonlinear Energy Transfers and Triple Correlations Global and Detailed Conservation Properties Advanced Analysis of Triadic Transfers and Waleffe's Instability Assumption Further Discussions About the Instability Assumption Principle of Quasi-Normal Closures EDQNM for Isotropic Turbulence. Final Equations and Results Topological Analysis, Coherent Events, and Related Dynamics 97

4 3.6.1 Topological Analysis of Isotropic Turbulence Vortex Tube: Statistical Properties and Dynamics Bridging with Turbulence Dynamics and Intermittency Nonlinear Dynamics in the Physical Space On Vortices, Scales, Wavenumbers, and Wave Vectors - What are the Small Scales? Is There an Energy Cascade in the Physical Space? Ill Self-Amplification of Velocity Gradients Non-Gaussianity and Depletion of Nonlinearity What are the Proper Features of Three-Dimensional Navier-Stokes Turbulence? Influence of the Space Dimension: Introduction to d-dimensional Turbulence Pure 2D Turbulence and Dual Cascade Role of Pressure: A View of Burgers' Turbulence Sensitivity with Respect to Energy-Pumping Process: Turbulence with Hyperviscosity 122 Bibliography Incompressible Homogeneous Anisotropic Turbulence: Pure Rotation Physical and Numerical Experiments Brief Review of Experiments, More or Less in the Configuration of Homogeneous Turbulence Governing Equations Generals Important Nondimensional Numbers. Particular Regimes Advanced Analysis of Energy Transfer by DNS Balance of RST Equations. A Case Without "Production." New Tensorial Modeling Inertial Waves. Linear Regime Analysis of Deterministic Solutions Analysis of Statistical Moments. Phase Mixing and Low-Dimensional Manifolds Nonlinear Theory and Modeling: Wave Turbulence and EDQNM Bull Exact Nonlinear Equations. Wave Turbulence Second-Order Statistics: Identification of Relevant Spectral-Transfer Terms Toward a Rational Closure with an EDQNM Model Recovering the Asymptotic Theory of Inertial Wave Turbulence Fundamental Issues: Solved and Open Questions Eventual Two-Dimensionalization or Not 153

5 4.7.2 Meaning of the Slow Manifold Are Present DNS and LES Useful for Theoretical Prediction? Is the Pure Linear Theory Relevant? Provisional Conclusions About Scaling Laws and Quantified Values of Key Descriptors Coherent Structures, Description, and Dynamics % 159 Bibliography Incompressible Homogeneous Anisotropic Turbulence: Strain Main Observations Experiments for Turbulence in the Presence of Mean Strain. Kinematics of the Mean Flow Pure Irrotational Strain, Planar Distortion Axisymmetric (Irrotational) Strain The Most General Case for 3D Irrotational Case More General Distortions. Kinematics of Rotational Mean Flows First Approach in Physical Space to Irrotational Mean Flows Governing Equations, RST Balance, and Single-Point Modeling General Assessment of RST Single-Point Closures Linear Response of Turbulence to Irrotational Mean Strain The Fundamentals of Homogeneous RDT Qualitative Trends Induced by the Green's Function Results at Very Short Times. Relevance at Large Elapsed Times Final RDT Results for Mean Irrotational Strain General RDT Solution Linear Response of Turbulence to Axisymmetric Strain First Step Toward a Nonlinear Approach Nonhomogeneous Flow Cases. Coherent Structures in Strained Homogeneous Turbulence 187 Bibliography Incompressible Homogeneous Anisotropic Turbulence: Pure Shear Physical and Numerical Experiments: Kinetic Energy, RST, Length Scales, Anisotropy Experimental and Numerical Realizations Main Observations 193

6 6.2 Reynolds Stress Tensor and Analysis of Related Equations Rapid Distortion Theory: Equations, Solutions, Algebraic Growth Some Properties of RDT Solutions Relevance of Homogeneous RDT Evidence and Uncertainties for Nonlinear Evolution: Kinetic-Energy Exponential Growth Using Spectral Theory Vortical-Structure Dynamics in Homogeneous Shear Turbulence Self-Sustaining Turbulent Cycle in Homogeneous Sheared Turbulence Self-Sustaining Processes in Nonhomogeneous Sheared Turbulence: Exact Coherent States and Traveling-Wave Solutions Local Isotropy in Homogeneous Shear Flows 214 Bibliography Incompressible Homogeneous Anisotropic Turbulence: Buoyancy and Stable Stratification Observations, Propagating and Nonpropagating Motion. Collapse of Vertical Motion and Layering Simplified Equations, Using Navier-Stokes and Boussinesq Approximations, With Uniform Density Gradient Reynolds Stress Equations With Additional Scalar Variance and Flux First Look at Gravity Waves Eigenmode Decomposition. Physical Interpretation The Toroidal Cascade as a Strong Nonlinear Mechanism Explaining the Layering The Viewpoint of Modeling and Theory: RDT, Wave Turbulence, EDQNM Coherent Structures: Dynamics and Scaling of the Layered Flow, "Pancake" Dynamics, Instabilities Simplified Scaling Laws Pancake Structures, Zig-Zag, and Kelvin-Helmholtz Instabilities 237 Bibliography Coupled Effects: Rotation, Stratification, Strain, and Shear Rotating^tratified Turbulence Basic Triadic Interaction for Quasi-Geostrophic Cascade About the Case With Small but Nonnegligible f/n Ratio 247

7 xii Contents The QG Model Revisited. Discussion Rotation or Stratification With Mean Shear The Rotating-Shear-Flow Case The Stratified-Shear-Flow Case Analogies and Differences Between the Two * Cases Shear, Rotation, and Stratification. RDT Approach to Baroclinic Instability Physical Context, the Mean Flow RDT Equations Elliptical Flow Instability From "Homogeneous" RDT Axisymmetric Strain With Rotation Relevance of RDT and WKB RDT Variants for Analysis of Classical Instabilities 266 Bibliography Compressible Homogeneous Isotropic Turbulence Introduction to Modal Decomposition of Turbulent Fluctuations Statement of the Problem Kovasznay's Linear Decomposition Weakly Nonlinear Corrected Kovasznay Decomposition Helmholtz Decomposition and Its Extension Bridging Between Kovasznay and Helmholtz Decomposition On the Feasibility of a Fully General Modal Decomposition Mean-Flow Equations, Reynolds Stress Tensor, and Energy Balance in Compressible Flows Arbitrary Flows Simplifications in the Isotropic Case Quasi-Isentropic Isotropic Turbulence: Physical and Spectral Descriptions Different Regimes in Compressible Turbulence Quasi-Isentropic Turbulent Regime Weakly Compressible Thermal Regime Nonlinear Subsonic Regime Supersonic Regime Structures in the Physical Space Turbulent Structures in Compressible Turbulence A Probabilistic Model for Shocklets 321 Bibliography 324

8 XIII 10 Compressible Homogeneous Anisotropic Turbulence Effects of Compressibility in Free-Shear Flows. Observations RST Equations and Single-Point Modeling Preliminary Linear Approach: Pressure-Released Limit and Irrotational Strain A General Quasi-Isentropic Approach to Homogeneous Compressible Shear Flows Governing Equations and Admissible Mean Flows Properties of Admissible Mean Flows Linear Response in Fourier Space. Governing Equations Incompressible Turbulence With Compressible Mean-Flow Effects: Compressed Turbulence Compressible Turbulence in the Presence of Pure Plane Shear Qualitative Results Discussion of Results Toward a Complete Linear Solution Perspectives and Open Issues Homogeneous Shear Flows Perspectives Toward Inhomogeneous Shear Flows Topological Analysis, Coherent Events and Related Dynamics Nonlinear Dynamics in the Subsonic Regime Topological Analysis of the Rate-of-Strain Tensor Vortices, Shocklets, and Dynamics 355 Bibliography Isotropic Turbulence-Shock Interaction Brief Survey of Existing Interaction Regimes Destructive Interactions Nondestructive Interactions Linear Nondestructive Interaction Shock Modeling and Jump Relations Introduction to the Linear Interaction Approximation Theory Vortical Turbulence-Shock Interaction Acoustic Turbulence-Shock Interaction Mixed Turbulence-Shock Interaction On the Use of RDT for Linear Nondestructive Interaction Modeling Nonlinear Nondestructive Interactions Turbulent Jump Conditions for the Mean Field Jump Conditions for an Incident Isotropic Turbulence 381 Bibliography 382

9 xiv Contents 12 Linear Interaction Approximation for Shock-Perturbation Interaction Shock Description and Emitted Fluctuating Field Calculation of Wave Vectors of Emitted Waves General Incident Entropy and Vorticity Waves Incident Acoustic Waves Calculation of Amplitude of Emitted Waves General Decompositions of the Perturbation Field Calculation of Amplitudes of Emitted Waves Reconstruction of the Second-Order Moments Case of a Single Incident Wave Case of an Incident Turbulent Isotropic Field A posteriori Assessment of LIA 403 Bibliography Linear Theories. From Rapid Distortion Theory to WKB Variants Rapid Distortion Theory for Homogeneous Turbulence Solutions for ODEs in Orthonormal Fixed Frames of Reference Using Solenoidal Modes for a Green's Function with a Minimal Number of Components Prediction of Statistical Quantities RDT for Two-Time Correlations Zonal RDT and Short-Wave Stability Analysis Irrotational Mean Flows Zonal Stability Analysis With Disturbances Localized Around Base-Flow Trajectories Using Characteristic Rays Related to Waves Instead of Trajectories Application to Statistical Modeling of Inhomogeneous Turbulence Transport Models Along Mean Trajectories Semiempirical Transport "Shell" Models Conclusions, Recent Perspectives Including Subgrid-Scale Dynamics Modeling 419 Bibliography Anisotropic Nonlinear Triadic Closures Canonical HIT, Dependence on the Eddy Damping for the Scaling of the Energy Spectrum in the Inertial Range 423

10 xv 14.2 Solving the Linear Operator to Account for Strong Anisotropy Random and Averaged Nonlinear Green's Functions Homogeneous Anisotropic Turbulence with a Mean Flow A General EDQN Closure. Different Levels of Markovianization EDQNM2 Version A Simplified Version: EDQNM The Most Sophisticated Version: EDQNM Application of Three Versions to the Rotating Turbulence Other Cases of Flows With and Without Production Effects of the Distorting Mean Flow Flows Without Production Combining Strong and Weak Turbulence Role of the Nonlinear Decorrelation Time Scale Connection with Self-Consistent Theories: Single Time or Two Time? Applications to Weak Anisotropy A Self-Consistent Representation of the Spectral Tensor for Weak Anisotropy Brief Discussion of Concepts, Results, and Open Issues Open Numerical Problems 446 Bibliography Conclusions and Perspectives Homogenization of Turbulence. Local or Global Homogeneity? Physical Space or Fourier Space? Linear Theory, "Homogeneous" RDT, WKB Variants, and LIA Multipoint Closures for Weak and Strong Turbulence The Wave-Turbulence Limit Coexistence of Weak and Strong Turbulence, With Interactions Revisiting Basic Assumptions in MPC Structure Formation, Structuring Effects, and Individual Coherent Structures Anisotropy Including Dimensionality, a Main Theme Deriving Practical Models 459 Bibliography 460 Index 461 *