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Modern Fluid Dynamics for Physics and Astrophysics

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Modern Fluid Dynamics for Physics and Astrophysics Oded Regev • Orkan M. Umurhan • Philip A. Yecko Modern Fluid Dynamics for Physics and Astrophysics & Springer Contents 1 Fundamentals 1 1.1 Continuum Description of Fluids 1 1.2 Kinematics of Fluid Motion 2 1.2.1 Lagrangian Description 3 1.2.2 Eulerian Description 6 1.2.3 Rate of Deformation and Rotation 8 1.3 Dynamics of Fluid Motion 9 1.3.1 Forces and Stresses 10 1.3.2 Cauchy Theory of Stress and Its Physical Meaning in a Fluid 11 1.3.3 Some Mathematical Relations 15 1.4 The Fluid Equations: Conservation Laws 18 1.4.1 Mass Conservation 18 1.4.2 Momentum Conservation 20 1.4.3 Energy Conservation 23 1.4.4 Summary of the Fluid Dynamical Equations 25 1.4.5 Examples 28 1.5 Thermodynamics of Fluid Motion 30 1.5.1 Local Thermodynamic Equilibrium 30 1.5.2 Equations of State and the Laws of Thermodynamics.... 31 1.6 Similarity and Self-Similarity in Fluid Dynamics 38 1.6.1 Similarity of Polytropic Stars Having the Same Index 41 1.6.2 The Self-Similar Solution of a Collapsing Isothermal Sphere 44 1.7 The Virial Theorem and Some of Its Consequences 46 1.7.1 General Derivation 46 1.7.2 Some Specific Consequences 49 xvii xviii Contents 2 Restricted and Vortical Flows 57 2.1 Introduction 57 2.2 Vorticity Basics and the Crocco Theorem 58 2.2.1 Crocco's Theorem 60 2.3 Some Basic Theorems and Results 61 2.3.1 Barotropic Flows 62 2.3.2 Inviscid Flows 62 2.3.3 Ertel's Theorem, Potential Vorticity 63 2.3.4 Kelvin's Theorem, Circulation 65 2.3.5 The Various Forms of Bernoulli's Theorem 67 2.3.6 Examples 69 2.4 Potential (Irrotational) Flows 75 2.4.1 Incompressible Potential Flows 80 2.4.2 General Three-Dimensional Potential Flow Past a Solid Body 82 2.4.3 Two-Dimensional Flows: Stream Function and Complex Potential 85 2.5 Vortex Motion 93 2.5.1 Helmholtz-Vortex Theorems 94 2.5.2 Inviscid Two-Dimensional Vortex Equation of Motion... 96 2.5.3 Hamiltonian Dynamics of Point Vortices 96 2.5.4 The Velocity Field, Derived from a Given Vorticity Field 98 2.5.5 The Rankine Vortex 99 2.5.6 Tumbling Kirchhoff-Kida Vortices 102 3 Viscous Flows 121 3.1 Introduction 121 3.2 Elementary Flows When the Governing Equations Are Linear.... 122 3.2.1 Plane Couette Flow 123 3.2.2 Poiseuille Flow 123 3.2.3 Flow on an Inclined Plane 125 3.2.4 The Rayleigh Problem 126 3.3 Some Additional Viscous Flows 128 3.3.1 Two-Dimensional Flow Towards a Stagnation Point 128 3.3.2 Flow in a Converging Channel 130 3.4 Motion in Very Viscous Fluid 131 3.4.1 Stokes Flow and Its Properties 132 3.4.2 The Stokes "paradox" and the Reciprocal Theorem 140 3.4.3 Viscosity of Suspensions 142 3.4.4 Hele-Shaw Flow 145 3.5 Viscous Boundary Layers 147 3.5.1 A Mathematical Digression 149 3.5.2 The Blasius Boundary Layer 152 3.5.3 Concluding Remarks on Boundary Layers 155 Contents xix 4 Linear and Nonlinear Incompressible Waves 161 4.1 Waves, A Mathematical Primer 162 4.1.1 One-Dimensional Linear Waves 166 4.1.2 One-Dimensional Linear Wave Equation Reexamined as an Initial Value Problem 168 4.1.3 Dispersion Relations, Phase and Group Velocities, Dirac Delta Function 174 4.1.4 One-Dimensional Unidirectional Nonlinear Waves and Their Breaking 182 4.2 Gravity Waves on Water Surface as Irrotational Flows 188 4.2.1 Formulation 188 4.2.2 Linearization and Waves: Scaling and Normal Mode Analysis 191 4.2.3 The Effect of Surface Tension 198 4.2.4 Surface Gravity Waves Induced by a Steady Flow Over a Corrugated Bed 202 4.3 Shallow Water Equations 205 4.3.1 Derivation via Scaling Analysis 206 4.4 Atmospheric Waves in the Boussinesq Approximation 211 4.4.1 The Boussinesq Approximation and Corresponding Equations 211 4.4.2 3-D Waves in Plane-Parallel Atmosphere 215 4.4.3 Energy Propagation in Internal Gravity Waves 218 4.5 Solitons in Shallow Water 221 4.5.1 An Asymptotic Derivation of the KdV Equation 223 4.5.2 Linear Theory Reanalyzed and Some Exact Solutions ... 229 5 Rotating Flows 241 5.1 Fundamentals 242 5.1.1 Fluid Equations of Motion in a Uniformly Rotating Reference Frame 243 5.1.2 Rossby, Burger, and Ekman Numbers 248 5.2 Some Rotating Flow Paradigms 251 5.2.1 Taylor-Proudman Theorem and Geostrophic Flow 251 5.2.2 Taylor-Couette Flow 254 5.2.3 Simple Ekman Layer 256 5.3 Linear Dynamics of Spin-Down 258 5.3.1 Scalings and Nondimensionalization 261 5.3.2 The Behavior in the Bulk-Interior: The Exterior Solution 262 5.3.3 Boundary Layers: One-Term Interior Solutions Near z = 0,1 264 5.3.4 Matching and Remarks 267 xx Contents 5.4 Linearized Dynamics: Inertial and Rossby Waves 269 5.4.1 Introduction to Inertia-Gravity Waves 269 5.4.2 Introduction to Rossby Waves 271 5.5 A Geophysical Example: Quasi-Geostrophy 273 5.5.1 Physical Assumptions 274 5.5.2 Expansion Procedure and Derivation 277 5.5.3 Conservation of Potential Vorticity and Some Remarks.. 279 5.5.4 Planetary Waves 282 5.5.5 Thermal Wind 284 5.6 An Astrophysical Example: Local Structure of Steady Thin Accretion Disks 286 5.6.1 The Vertically Averaged Steady Model 289 5.6.2 The Asymptotic Polytropic Model with Vertical Structure 293 5.6.3 Summary 295 6 Effects of Compressibility 305 6.1 Introduction 305 6.1.1 A Historical Note 306 6.1.2 Overview of This Chapter 308 6.2 Sound 309 6.2.1 The Acoustic Wave Equation 310 6.2.2 Plane Sound Waves 315 6.2.3 Spherical Sound Waves 319 6.2.4 Energy and Momentum Transport in Acoustic Waves.... 323 6.2.5 Normal Modes of Acoustic Vibrations 327 6.2.6 The Emission and Attenuation of Sound 331 6.3 Properties of Compressible Flows 341 6.3.1 Propagation of Disturbances, the Mach Cone 342 6.3.2 Compressible Potential Flow 346 6.3.3 Isentropic Flow of a Compressible Perfect Gas 348 6.4 One-dimensional Gas Dynamics 353 6.4.1 Characteristics 353 6.4.2 Riemann Invariants, the Domains of Determination and of Influence 359 6.4.3 Nonlinear Simple Waves and the Steepening of Such Waves 362 6.4.4 Rarefaction Waves 365 6.4.5 The Mathematical Analogy with Shallow Water Waves and Its Consequences 368 6.4.6 Examples of Additional Specific Topics 375 6.5 ShockWaves 378 6.5.1 General Shock Conditions 378 6.5.2 Rankine-Hugoniot Adiabat and Jump Conditions 382 6.5.3 Radiative Shocks 389 Contents XX1 6.6 Explosions, Blast, and Detonation Waves 390 6.6.1 Strong Explosion at a Point: Blast Wave 390 6.6.2 Detonation Waves 395 7 Hydrodynamic Stability 409 7.1 Introduction: The Experiments of Reynolds and Taylor 409 7.2 Fundamentals of Linear Stability Theory 414 7.3 Stability of Plane-Parallel Shear Flows 420 7.3.1 Three-Dimensional Disturbances in Shear Flows 420 7.3.2 Inviscid Shear 424 7.3.3 Viscous Shear 430 7.4 Buoyant Instabilities 434 7.4.1 Rayleigh-Taylor Instability 435 7.4.2 Rayleigh-Benard Convection 440 7.4.3 Taylor-Couette Centrifugal Instability 449 7.5 Additional Instabilities 454 7.5.1 Rayleigh-Plateau-Savart Breakup of a Liquid Jet 455 7.5.2 Jeans Instability 457 7.6 Instability Due to Transient Growth 461 7.6.1 Pipe Poiseuille Flow and the Failure of Normal Mode Analysis 461 7.6.2 Energy Growth Due to Non-normality 463 8 Weakly Nonlinear Instability 473 8.1 Introduction 473 8.2 Amplitude Equations and Multiple Scale Analysis 475 8.2.1 Multiple Scale Analysis Near Onset 478 8.3 Stationary Rolls in Rayleigh-Benard Convection I: Fixed Temperature 481 8.3.1 The Amplitude Equation 484 8.3.2 Patterns in Three Dimensions 490 8.4 Stationary Rolls in Rayleigh-Benard Convection II: Fixed Flux 493 8.5 Additional Systems 496 8.5.1 Subcritical Transitions 496 8.5.2 Faraday Instabilities 497 8.5.3 Model Equations 499 8.6 Beyond Weak Nonlinearity 501 9 Turbulence 507 9.1 Introduction 507 9.1.1 The Origin of Turbulence, Its Ubiquity, and Some Basic Properties 509 9.1.2 Topics Discussed in This Chapter 512 xxii Contents 9.2 Reynolds Equations and Shear Flows 513 9.2.1 Reynolds Stress and the Closure Problem 514 9.2.2 Energy Transfer from the Mean Flow to the Turbulence . 519 9.2.3 Wall-Bounded Flows and the Log-Law 522 9.2.4 Free Shear Flows 526 9.3 Rudiments of a Statistical Description of Turbulence 530 9.3.1 Averaging 530 9.3.2 Correlation Functions, Velocity Increments, and Structure Functions 533 9.3.3 Random Fields 539 9.3.4 Sketch of the Proof of the Ergodic Theorem in Turbulence Theory 542 9.3.5 The Statistical Formulation of the Fundamental Problem of Turbulence 544 9.4 Some Theory, Mostly Kolmogorov 546 9.4.1 Introduction 546 9.4.2 A Theoretical Simplification for Periodic Boundary Conditions 549 9.4.3 Two Experimental Findings 555 9.4.4 Rudiments of Kolmogorov's 1941 Theory 558 9.4.5 Phenomenology Revisited, in View of Kolmogorov's Work 563 9.5 Two-Dimensional Turbulence 568 9.5.1 Governing Equations and Integral Constraints 570 9.5.2 Inverse Energy Cascade 573 9.5.3 Direct Enstrophy Cascade 577 9.5.4 Freely Decaying Two-Dimensional Turbulence 578 9.5.5 Experiments and Simulations 580 9.6 A Few Remarks on Turbulence Is Astrophysical Flows 580 9.6.1 Turbulent Convection 580 9.6.2 More Turbulence in Astrophysics 583 9.7 Very Brief Summaries on a Few Additional Topics 583 9.7.1 Intermittency 584 9.7.2 Dynamical Systems Approach to Turbulence 585 9.7.3 Two Basic Approaches to Simulation 585 10 Magnetohydrodynamics 595 10.1 Introduction 595 10.1.1 Estimates of Some Kinetic Coefficients 599 10.2 MHD One-Fluid Model Equations 602 10.2.1 Combining the Maxwell Equations with the FD Equations 602 10.2.2 The Induction Equation and the Fluid Heat Equations in MHD 606 10.2.3 Flux Tubes and Their Properties 612 Contents xxiii 10.3 Equilibrium Configurations 614 10.3.1 Example of Cylindrical Unbounded Configuration (Pinch) 615 10.3.2 The Grad-Shafranov Equation 616 10.3.3 MHD Equilibria in a Gravitational Field 618 10.3.4 Force-Free Fields 619 10.4 MHD Waves 620 10.4.1 Magnetic Waves 622 10.4.2 Magneto-acoustic Waves 625 10.5 Discontinuities and MHD Shock Waves 626 10.5.1 MHD Rotational (Alfven) Discontinuities 628 10.5.2 MHD Shock Waves 629 10.6 Some Common MHD Instabilities 631 10.6.1 Example: Parker Buoyancy Instability, Interstellar Clouds, and Sunspots 637 10.7 Rudiments of Dynamo Theory 639 10.8 Short Overviews of Two Real Research Problems 647 10.8.1 Fusion Research 647 10.8.2 Astrophysical Jets 648 10.9 Further Reading on Three Additional Topics 650 10.9.1 MHD Turbulence 650 10.9.2 Reconnection 650 10.9.3 Hall MHD 651 A Vector Formulae 663 A.l Identities 663 A.
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