Vortical Flows

This book is a comprehensive and intensive book for graduate students in fluid dynamics as well as scientists, engineers and applied mathematicians. Offering a systematic introduction to the physical theory of vortical flows at graduate level, it considers the theory of vortical flows as a branch of fluid dynamics focusing on __shearing process__ in fluid motion, measured by __vorticity__. It studies vortical flows according to their natural evolution stages,from being generated to dissipated. As preparation, the first three chapters of the book provide background knowledge for entering vortical flows. The rest of the book deals with vortices and vortical flows, following their natural evolution stages. Of various vortices the primary form is __layer-like vortices__ or __shear layers__, and secondary but stronger form is __axial vortices__ mainly formed by the rolling up of shear layers. Problems are given at the end of each chapter and Appendix, some for helping understanding the basic theories, and some involving specific applications; but the emphasis of both is always on physical thinking. Preface 5 Acknowledgments 7 Contents 8 1 Fundamentals of Fluid Dynamics 14 1.1 Basic Fluid Kinematics 14 1.1.1 Description and Visualization of Fluid Motion 14 1.1.2 Dilatation and Vorticity 20 1.1.3 Velocity Gradient and Its Decompositions 22 1.1.4 Local and Global Material Derivatives 28 1.2 Dynamic Equations of Fluid Motion 32 1.2.1 Dynamic Equations for General Fluids 32 1.2.2 Constitutive Relations and Thermodynamics 35 1.2.3 Navier-Stokes Equations and Perfect Gas 40 1.2.4 Dominant Non-dimensional Parameters 42 1.3 Wall-Bounded Flows 45 1.3.1 Boundary Conditions 45 1.3.2 Fluid Reaction to Solid Boundaries 46 1.4 Problems for Chapter 1 49 2 Fundamental Processes in Fluid Motion 52 2.1 Preliminary Observations 52 2.2 Intrinsic Decomposition of Fundamental Processes 55 2.2.1 Helmholtz Decomposition 55 2.2.2 Dynamic Equations for Vorticity and Dilatation 57 2.3 Coupling and Splitting of Fundamental Processes 60 2.3.1 Process Nonlinearity and Coupling Inside the Flow 61 2.3.2 Process Linear Coupling on Boundaries 63 2.3.3 Linearized Process Splitting in Unbounded Space 67 2.4 Far-Field Asymptotics in Unbounded Flow 69 2.4.1 Vorticity and Dilatation Far Fields 69 2.4.2 Velocity Far Field 71 2.4.3 Far-Field Asymptotics for Steady Flow 74 2.5 A Decoupled Model Flow: Inviscid Gas Dynamics 77 2.5.1 Basic Equations 77 2.5.2 Unsteady Potential Flows 78 2.5.3 Steady Isentropic Flow 79 2.6 Minimally-Coupled Model: Incompressible Flow 80 2.6.1 Momentum Formulation versus Vorticity Formulation 80 2.6.2 Incompressible Potential Flow 83 2.6.3 Accelerated Body Motion and Virtual Mass 86 2.6.4 Force on a Body in Steady Flow 87 2.7 Problems for Chapter 2 88 3 Vorticity Dynamics 90 3.1 Kinematic Properties of Vorticity Field 90 3.1.1 Vorticity Tube and Circulation 90 3.1.2 Geometric Relation of Velocity and Vorticity 93 3.1.3 Two-Dimensional and Axisymmetric Vortical Flows 99 3.1.4 Biot-Savart Formulas 101 3.2 Vorticity Kinetic Vector and Circulation-Preserving Flow 106 3.2.1 General Evolution Formulas 106 3.2.2 Local Material Invariants 108 3.2.3 Vorticity-Tube Stretching and Tilting 111 3.2.4 Bernoulli Integrals 113 3.3 Vorticity Integrals and Their Invariance 116 3.3.1 Total Vorticity and Circulation 117 3.3.2 Lamb-Vector Integrals 118 3.3.3 Vortical and Potential Impulses 120 3.3.4 Helicity 126 3.3.5 Total Kinetic Energy 128 3.4 Physical Causes of Vorticity Kinetics 130 3.4.1 Coriolis Force in Rotating Fluid 131 3.4.2 Baroclinicity 133 3.4.3 Vorticity Diffusion and Enstrophy Dissipation 136 3.4.4 Vorticity Creation at Boundary 138 3.5 Problems for Chapter 3 141 4 Attached and Free Vortex Layers 147 4.1 Parallel Shear Flows on Upper-Half Plane 147 4.1.1 General Solution in Vorticity Formulation 148 4.1.2 Singular BVF: Stokes First Problem (Rayleigh Problem) 150 4.1.3 Oscillatory BVF: Stokes Second Problem 151 4.2 Boundary Layers: Formulation and Physics 153 4.2.1 From d'Alembert's Paradox to Prandtl's Theory 153 4.2.2 From Rayleigh Problem to Boundary Layer Equations 156 4.2.3 Blasius Boundary Layers 159 4.2.4 Further Issues 160 4.2.5 Vorticity Dynamics in Boundary Layer 164 4.3 High-Frequency Oscillatory Boundary Layer 167 4.4 Free Steady Vortex Layers 170 4.4.1 Free Shear Layer 170 4.4.2 Jet 171 4.4.3 Far Wakes 175 4.5 Problems for Chapter 4 177 5 Vortex Sheet Dynamics 179 5.1 Basic Properties of Free Vortex Sheet 179 5.1.1 Strength and Velocity of Free Vortex Sheet 180 5.1.2 Circulation, Lamb Vector, and Bernoulli Equation 181 5.2 Attached Vortex Sheet and Its Separation 183 5.2.1 Attached and Bound Vortex Sheet 183 5.2.2 Kutta Condition and Vortex-Sheet Separation 186 5.3 Motion of Free Vortex Sheet 189 5.3.1 Rolling up and Kaden's Similarity Law 189 5.3.2 Methods of Computing Vortex Sheet Motion 193 5.4 Formation of Wing Vortices 194 5.4.1 Formation of Wingtip Vortices 194 5.4.2 Formation of Leading-Edge Vortex 196 5.5 On the Role of Vortex-Sheet Dynamics 197 5.6 Problems for Chapter 5 200 6 Axisymmetric Columnar Vortices 202 6.1 General Background 203 6.1.1 Governing Equations and Their Simplifications 203 6.1.2 Simplified Axisymmetric Model Equations 205 6.2 Two-Dimensional Stretch-Free Vortices 206 6.2.1 Steady and Inviscid Pure Vortices 206 6.2.2 Unsteady and Viscous Pure Vortices 207 6.3 Radial-Axial Flow Coupling and Stretched Vortices 209 6.3.1 Burgers Vortex 210 6.3.2 Sullivan Vortex 211 6.4 Azimuthal-Axial Flow Coupling and Batchelor Vortex 213 6.4.1 Slender and Light-Loading Approximation 214 6.4.2 Azimuthal-Axial Flow Coupling 215 6.4.3 Batchelor Vortex 216 6.5 Trailing Vortex with Composite Core Structure 218 6.5.1 Composite Core Structure 219 6.5.2 Moore-Saffman Trailing Vortex 220 6.6 Problems for Chapter 6 222 7 Vortex Rings 225 7.1 General Formulation and Properties 225 7.1.1 Governing Equations 225 7.1.2 Integral Invariants 229 7.1.3 Stokes Streamfunction 230 7.2 Inviscid Vortex Rings 232 7.2.1 Thin-Core Vortex Ring 232 7.2.2 Hill's Spherical Vortex 235 7.2.3 Fraenkel-Norbury Vortex Ring Family 238 7.3 Evolution of Viscous Vortex Rings 240 7.3.1 Early Stage at νT/R20 ll1 240 7.3.2 Matured Stage at νT/R20 =O(1) 242 7.3.3 Late Stage at νT/R20gg1 243 7.4 Problems for Chapter 7 245 8 Flow Separation and Separated Flows 247 8.1 Orientation 247 8.2 Generic Steady Flow Separation 249 8.2.1 Separation Criteria 249 8.2.2 Dynamic System and Fixed Points 253 8.2.3 Near-Wall Dynamic System for Flow Separation 256 8.3 Steady Boundary-Layer Separation 259 8.3.1 Deck Structure and Scale Analysis 260 8.3.2 Triple-Deck Equations and Self-induced Pressure Gradient 262 8.3.3 Three-Dimensional Triple Deck 264 8.4 Steady Separated Flows 266 8.4.1 Steady Separated Bubble Flow 266 8.4.2 Fixed-Point Index and Topology of Vector Field 269 8.4.3 Topological Diagnosis of Separated Flows 273 8.4.4 Structural Stability 274 8.4.5 Open Separation with Boundary-Layer Breaking Away 277 8.5 Unsteady Separation and Separated Flow 278 8.5.1 A Highlight of Unsteady Separation 279 8.5.2 Formation of Airfoil Circulation in Starting Flow 280 8.5.3 Separated Flow Over Circular Cylinder 284 8.5.4 Falling Disk in Still Water 287 8.6 Problems for Chapter 8 290 9 Vortical Fluid-Dynamic Force and Moment 292 9.1 Origin of Lift 293 9.1.1 Inviscid Circulation Theory and Criticisms 294 9.1.2 Viscous Circulation Theory 298 9.1.3 Further Issues 302 9.2 Classic Steady Vortical Aerodynamics 304 9.2.1 Steady Lift on Airfoil 304 9.2.2 Steady Lifting-Line Theory 308 9.3 Classic Unsteady Vortical Aerodynamics 312 9.3.1 Vortical Impulse Theory 312 9.3.2 Force and Moment on Unsteady Thin Airfoil 314 9.4 A General Formulation of Vortical-Force Theory 321 9.4.1 Pressure Removal 321 9.4.2 Advection Form of Vortical Force 322 9.4.3 Diffusion Form of Vortical Force 326 9.4.4 Boundary Form of Vortical Force 327 9.5 Problems for Chapter 9 329 10 Vortex Instability, Breakdown, and Transition to Turbulence 333 10.1 Basic Concepts of Vortical-Flow Stability 333 10.1.1 Normal-Mode Analysis 334 10.1.2 Nonmodal Analysis and Transient Growth 338 10.1.3 Receptivity 340 10.2 Instability of Axisymmetric Columnar Vortices 340 10.2.1 Stability of Pure Vortices 341 10.2.2 Temporal Instability of Swirling Vortices 342 10.2.3 Absolute and Convective Instability of Swirling Vortices 345 10.2.4 Instability of Trailing Vortex Pair 348 10.3 Vortex Breakdown 352 10.3.1 Breakdown in Terms of Vorticity Dynamics 354 10.3.2 Breakdown in Term of AI/CI 356 10.4 Vortex Ring Instability and Transition 357 10.4.1 Linear Instability: Single Vortex Ring 358 10.4.2 Nonlinear Instability and Transition: Single Vortex Ring 359 10.4.3 Instability and Transition: Multiple Vortex Rings 360 10.5 Problems for Chapter10 362 Problems for Chap.10 362 11 Vortical Structures in Transitional and Turbulent Shear Flows 368 11.1 Overview and Background 368 11.1.1 What Is Turbulence 368 11.1.2 Mean Turbulent Flow 373 11.1.3 Vorticity Equations and Statistics 377 11.2 Instability and Transition of Free Shear Layer 379 11.2.1 Instability of Free Shear Layer 379 11.2.2 Free and Forced Evolutions of Spanwise Vortices 381 11.2.3 Secondary Instability and Formation of Streamwise Vortices 386 11.2.4 Vortex Interaction and Small-Scale Transition 387 11.3 Instability and Transition of Wall Shear Layer 389 11.3.1 Instability Waves and Coherent Structures 389 11.3.2 Secondary Instability and Self-sustaining Cycle of Wall Turbulence 392 11.3.3 Transient Growth and Bypass Transition 394 11.3.4 Hairpin Vortices and Hairpin-Vortex Packets 396 11.3.5 Hypersonic Boundary-Layer Instability and Transition 402 11.4 Two Basic Physical Processes in Turbulence 407 Appendix: Fields of Vectors and Tensors 412 References 438 Index 448 This book is a comprehensive and intensive book for graduate students in fluid dynamics as well as scientists, engineers and applied mathematicians. Offering a systematic introduction to the physical theory of vortical flows at graduate level, it considers the theory of vortical flows as a branch of fluid dynamics focusing on shearing process in fluid motion, measured by vorticity. It studies vortical flows according to their natural evolution stages,from being generated to dissipated. As preparation, the first three chapters of the book provide background knowledge for entering vortical flows. The rest of the book deals with vortices and vortical flows, following their natural evolution stages. Of various vortices the primary form is layer-like vortices or shear layers, and secondary but stronger form is axial vortices mainly formed by the rolling up of shear layers. ℗lProblems are given at the end of each chapter and Appendix, some for helping understanding the basic theories, and some involving specific applications; but the emphasis of both is always on physical thinking This book is a comprehensive and intensive book for graduate students in fluid dynamics as well as scientists, engineers and applied mathematicians. Offering a systematic introduction to the physical theory of vortical flows at graduate level, it considers the theory of vortical flows as a branch of fluid dynamics focusing on shearing process in fluid motion, measured by vorticity. It studies vortical flows according to their natural evolution stages, from being generated to dissipated. As preparation, the first three chapters of the book provide background knowledge for entering vortical flows. The rest of the book deals with vortices and vortical flows, following their natural evolution stages. Of various vortices the primary form is layer-like vortices or shear layers, and secondary but stronger form is axial vortices mainly formed by the rolling up of shear layers. Problems are given at the end of each chapter and Appendix, some for helping understanding the basic theories, and some involving specific applications; but the emphasis of both is always on physical thinking Front Matter....Pages i-xiv Fundamentals of Fluid Dynamics....Pages 1-38 Fundamental Processes in Fluid Motion....Pages 39-76 Vorticity Dynamics....Pages 77-133 Attached and Free Vortex Layers....Pages 135-166 Vortex Sheet Dynamics....Pages 167-189 Axisymmetric Columnar Vortices....Pages 191-213 Vortex Rings....Pages 215-236 Flow Separation and Separated Flows....Pages 237-281 Vortical Fluid-Dynamic Force and Moment....Pages 283-323 Vortex Instability, Breakdown, and Transition to Turbulence....Pages 325-359 Vortical Structures in Transitional and Turbulent Shear Flows....Pages 361-404 Back Matter....Pages 405-446